I would absolutely continue with Paladin.
Aura of Protection is, like, the definitive paladin ability; it helps you and the whole party significantly. I'd never want to build a Paladin without at least the hope of getting to 6th.
After that, rogue gets you better mobility, a skill and expertise (more skills are always good); you'd need to get to Bard 2 to get the expertise bonus, but it'd probably be worth it for spells.
But then, consider Paladin 7 -- relentless avenger gives you a sweet mobility boost for combat (and then the ASI at 8th...)
Although I've been ruminating on many other thoughts, I really do have to second this. Aura of Protection is an
extremely useful, and (IMO) iconic, Paladin ability.
However, if you have your heart set on blending in other classes, either immediately or in the future, you have a few good options:
1. Sorcerer.
Personally, I think this makes by far the most sense, of the choices available to you. You've literally just slain a dragon--the dragon you were trying to slay. You have basked in its flames(/ice/etc. since you didn't say the color
) and lived. You have smelled--perhaps even tasted--its blood. Its magic is
part of you now, whether you like it or not. Or, at least, that's one possible way you could fluff the development of draconic Sorcerer-powers. You've got generally strong stats, and already being a Paladin means you've got that sweet, sweet heavy armor proficiency already locked in--remember that you can cast
any spells in armor you're proficient with. Sorcerer would be a good way to make up for the two major "holes" in the Paladin's arsenal: long-ranged and area-of-effect damage. You already kick major butt in melee thanks to your feats and good stats, and Paladin gives you a load of excellent support abilities. On top of that, just like the Paladin, I don't think there are any substantial Sorcerer abilities that depend on anything but Charisma, numerically, so the classes have maximal synergy. I don't know quite how Sorcery points would interface with your Paladin spells, but I'd assume it would only help, rather than hinder--you'll be a versatile, supportive killing machine, which is a hard combo to top in combat.
In summary: You've got a rock-solid flavor reason (I just killed a dragon and its power/spirit/blood now lives on in me), and tantalizing mechanical benefits that synergize well with your base class. On the downside, Sorcerer doesn't give you a lot to work with in terms of melee; it's mostly a source of extra spell slots and interesting spells known, rather than specifically empowering your melee abilities.
2. Warlock.
Flavor-wise, I think Warlock is just a single step down from Dragon Sorcerer. You've already said that your character has been getting power from
somewhere and just hasn't really stopped to question it until now. What if that power is "really" from a Warlock patron, who's been giving your character these abilities "pro bono" to see if he's worth a full-on "endorsement deal" kind of thing? This could even be, say, a spirit or deity of (good) dragons, who's tired of playing the "good parent" and decided the dragons need some tough love? "These vagabonds have gotten lazy, arrogant, and stupid. They no longer deserve the power I gave them--but you've proven you do. Remind my children that they still answer to me, and rewards--from me and their ill-gotten hoards--will be yours."
Although you're eyeing Blade Pact, I think it might be worth considering Tome as well. Blade doesn't give you as much as it could, as you've already got Extra Attack, while Tome opens up a
wealth of interesting non-combat magic in the form of ritual casting. With the recent Sage Advice
errata "clarification," you won't be able to get Lifedrinker (the key Blade Pact Invocation) until you take 12 levels of Warlock--or, in other words, 17th level at the earliest. If your DM ignores that particular "clarification," though, you could get it at character level 12, and it would synergize quite well with Improved Divine Smite (making your normal attacks deal 2d6+1d8+3(str)+2(cha)+10(GWM), which is frankly ridiculous; then on top of that, the normal Divine Smite can be used with your Warlock slots, which gives you a
substantial boost to daily damage output. And regardless of your pact, you can pick up Eldritch Blast and an invocation or two to make it awesomer: I strongly recommend both the damage booster (Agonizing Blast) and the knockback (Repelling Blast), so you can emulate FUS RO DAH! Alternatively, if you aren't already proficient with Deception and Persuasion, that's a
great Invocation choice, giving you effectively 2/3 of a feat as a class feature. Although a ~3 level dip would probably be the sweetspot, going for a full 5 levels gets you an ASI, two 3rd level slots (= +4d8 damage twice per short rest!), and a third invocation.
In summary: Not as far-fetched as you might think, given your statement that the character has already been getting power
from somewhere. Just as nicely synergized with Paladin as Sorcerer, but with a slightly trickier bend to it; the biggest benefits will be in branching out to new and especially non-combat options. Slots per short rest are excellent, but most of the Blade Pact features aren't really useful to you anymore, especially what with the "Warlock level, not character level, prerequisite" rules change/"clarification." Something of a flavor downside is that none of the patrons really smack of Dragon-y or anti-Dragon-y stuff; it's all too orthogonal.
3. Valor Bard
This one I'm having the hardest time justifying. Again, like the previous two, it's got great mechanical synergy, but I just can't quite see how Bard is related to being a dragonslayer or having just slain a dragon. I suppose you could feel super inspired by having achieved your current life goal? I dunno. Or you could just decide, "hey, I need to DO something with my life, now that I've accomplished my vengeance!" I leave those considerations up to you.
Unfortunately, Valor Bard has almost nothing to offer you. You already have Martial weapons, Medium Armor, and Shield proficiencies due to being a Paladin--so 3rd level really loses a lot of its shine. Battle Inspiration is still good, enhancing your ability to support your allies...but using Bardic Inspiration costs your bonus action, and that's already highly coveted for casting Smite spells, so it's decidedly un-ideal. 6th level is even worse, as you get practically nothing at all--Countercharm costs your action, making it a huge waste as a heavy-bruiser type, and it'll be half useless anyway because at Paladin 10 you get Aura of Courage which makes allies within 10 feet *immune* to fear! And finally, Battle Magic means not only going pure Bard basically from here on out, it also costs your action and doesn't play nice with Extra Attack nor your bonus-action Smite spells (though, admittedly, it does synergize with Divine Smite and Improved Divine Smite).
On the other hand, Lore Bard could give you quite a lot. You get
three bonus skill proficiencies at level 3 (the equivalent of a free feat), on top of having the generic Bard Jack of All Trades and Expertise effects (for a whopping total, presumably, of 8-9 total skill proficiencies, and by char level 9, a +2 to all untrained checks, which you're already good at because of your high stats!) I dunno if you have much call for using your reaction right now, so Cutting Words could be extremely useful or merely creating new competition for your reaction--but it's a very strong, useful ability. And of course extra Magical Secrets is
always great, stealing awesome spells from other classes is a great thing no matter what class you are.
In summary: Any form of Bard works with Paladin...but Valor Bard would probably be a waste of time, giving you stuff you literally can't benefit from. Lore Bard, on the other hand, gives you a great enemy debuff, a buttload of extra skills, and greater flexibility in the long run. On the downside, though, we have little to no reason (based on what you've said) for
why you'd become a Lore Bard and pick up a bunch of extra skills. Perhaps the dragon's ancient knowledge rubbed off on you? Heh.
Having written all of that now, it seems pretty clear: there are no classes you can MC into that will make you better at mauling things in the face. You're already in one of the best meleeist-with-spells classes there is; anywhere else you go will be, at best, maintaining parity, and at worst a waste of your time. If you're really serious about multiclassing, you'll need to look at it a completely different way. That is, think of it as, "I'm
already super good at smashing things in the face For Justice!, what
else could I be doing on top of that?" Sorcerer gives you raw damage + metamagic-ing your Divine spells; Warlock gives you Invocations, a Pact and a Patron, and per-short-rest spells; Lore Bard gives you oodles of skill things, enemy debuffing, and wider spell selection. If one of those things strikes you as being particularly cool, go for it; if not, pure Paladin is a perfectly cromulent choice, though I entirely understand if you feel a little..."bored" with it, if you know what I mean. (Paladin is my favorite class archetype...I just find the 5e version painfully dull, from a purely mechanical standpoint; powerful, sure, and the Oaths are a neat idea, but the mechanical execution is
so dull.)
