tl;dr: How do I handle a fallen paladin?
I have a level 6 player who wanted to play a lawful good aasimar paladin (Oath of Devotion). There have been several events which I believe have led to their fallen state.
The player agreed that they're ok with not being a paladin anymore. So I think that most paladins can either choose a new class or become an oathbreaker, if they break their oath. This character does not want to become an oathbreaker. They want to take a new class. The player doesn't want to change characters; I already suggested that.
How should I handle a paladin who wants to take on a new class?
For example, should they lose all their paladin benefits (skills, proficiencies, spells, etc.) and come in as a level 1 character of the new class? Or should they come in as a level 6 character (their current level) of the new class? Or should they just be a level 6 paladin who can't do magic, and then they are forced to multiclass?
This is also a notoriously stubborn player who likes to disregard rules that they don't like and make up new rules that fit them. But I also can't kick them from the campaign so I'm trying to be flexible.
Best Answer
There are (currently) no rules in any 5e book for retraining or changing any class features that a character already has. This means that you're in very definite make-it-up-as-you-go territory.
What To Do
Work with this player. Talk to your player and make sure they're OK with you changing their character. Listen to their ideas about how it could fit into the story and become part of their character's development and growth.
Be fair to your other players. Be open with the other players in that game so that they know what's happening, and make it clear that this is just how you're running the game, and not any kind of pandering to this player who likes to bend the rules. Let them know that the same sort of thing is available to them, if they wish.
Let them rebuild in whichever way fits the theme. Work out what they want from their class. Do they want something similar to a paladin? Do they want more martial? More magic? A different playstyle? Look at the other classes to see which ones provide options that interest and excite this player. When you find something they're happy with, simply swap the character to a different class at the same level and XP as they were. Basically, you're building a new character, but using the same name and backstory - and probably the same ability scores, but that's up to you.
Consider all the options. In particular, decide whether you're OK with multiclassing. Multiclassing lets you build more complex and varied characters, but it's also harder to keep a track of, and there are numerous pitfalls with it that result in the character being significantly less powerful than the average single-class character of equivalent level. Decide whether both DM and player are OK with the increased complexity, and if so, decide whether you're confident that you can competently build a character that keeps pace with the power level of the other characters in the party (or else that you don't mind this character being a little weaker, perhaps because you're more of a roleplay-heavy group).
Also think about letting them change their ASIs (ability score improvements) and feats around a little to help with the flavour - perhaps in their old build, they chose a +2 to strength, but now they want to retain a little bit of the paladin feeling, so you let them drop that +2 and take Ritual Caster instead, picking up a few cleric ritual spells for that divine-magic-user feeling. You can explain it away as the ASI being their Oath empowering them, and without the Oath, they are physically weakened (or whatever).
What Not To Do
Don't make them start at level 1 again. A 1st-level character cannot keep up with a 6th-level party. Either they'll spend every combat hiding and fleeing, or they'll be continuously knocked down until misfortune and unlucky dice rolls kills them off. Any encounter that will challenge a party of three-to-six 6th-level adventurers is quite capable of outright killing a 1st-level adventurer in a single blow... at which point they have to start again as another 1st level character, who will die even faster as the rest of the party continues to level up and seek tougher challenges.
Don't force them to change characters. Their current character has shared experiences with the party, has gained their trust, met their NPC acquaintances, and become invested in the plot. A new character has to do all that again from scratch, which means that this player might be sidelined for a while (because they're not invested in what's happening) or might hog the spotlight for a while (because you have to do things to bring them into the party properly), and you don't want either of those. There are also all sorts of loose ends left when a character exits - quests they'll never complete, backstory hooks they won't get to explore, promises made to NPCs that won't be kept, etc. Much better to keep the character around so that their story can continue.
Don't just take away some of their abilities and make them keep playing like that. This will leave them with a character who basically can't do anything interesting, which is very boring, and also means that they'll be significantly weakened and at risk of dying in combat. Even if they can start taking levels in another class from here onwards, multiclass builds are often weaker even when you have full access to both classes; taking away their paladin features and making them multiclass from here onward would basically put them 6 levels behind the power level of the party on all measures except hit points.
Don't let the player force you into something more complex than you're comfortable with. Multiclassing adds complexity, as do feats - there's a reason that these are both optional rules, and it's the DM who decides whether to use them. Similarly, don't let them push you into letting them use Unearthed Arcana or homebrew unless you're absolutely sure you're comfortable having those things in your game, and you're confident in your ability to keep things balanced.
Don't let frequent character rebuilds become the norm. As much as it's nice to let players rebuild their character in response to big story events that have a profound effect on them, do remember that events of this magnitude are rare - and that frequently rebuilding characters like this is going to take a lot of time and effort, and will seriously complicate your life as DM. It's hard enough already to keep track of all your player characters and what they can do; if your players can turn up with a new character build every week then it becomes near impossible.
Anecdotal Case Study
I had this same situation happen to me: I was playing an 8th-level Paladin who discovered that they had been thoroughly manipulated, and what our party had been doing was actually helping the villain. My oath was a large part of what had convinced me to do what we'd been doing, so in anger at being deceived, I renounced the oath and chose another path. After a chat with my DM, I rebuilt the character with two levels of Paladin and six of Warlock, taking Pact of the Blade, and the Fiend as my patron. Warlocks use Charisma for their spells, and so do Paladins; I had access to the Fighting Style and the paladin's armour and weapon proficiencies, which makes sense, because those things come from training and you'd probably remember them. I still had a handful of 1st-level Paladin spells, as well as Divine Smite, which left me with a little bit of that flavour of a former paladin.
However, I lost access to Channel Divinity, which is key, because that's very much linked to the Oath and its power. Instead, I had Warlock's Pact Magic, choosing spells that fit a fire-and-brimstone theme to show that anger that led me to renounce the Oath. I also found that the spell vampiric touch, as well as the feature Dark One’s Blessing (gain temporary hit points on slaying an enemy) are both great options for a character who learned how to Lay On Hands before leaving the path of pure good and maybe starting to think about ways that the manipulation of life force could be applied offensively.
Conclusion
As with many things in D&D, this is something for you to decide yourself, and as a result, your best option is to talk to the player in question - and probably the others too - and figure out what works best for the kind of game you want to play.