RAW
From strict reading of the ASI, you shouldn't be able to, since it does not say "You can't increase an ability score above your current cap" or something on these lines, it just says "You can't increase [...] above 20".
Rules as (probably) Intended
Now, is it intended? Probably not.
In the PHB, the only way to go over your cap is through Barbarian's Primal Champion feature, which is at 20th level and you are not getting any more ASIs after that.
So one possible reading on this is that it's worded as it is for simplicity, since they were not considering the items on DMG.
The problem is that, from my experience, the DMG is written (more) vaguely (than the PHB), which might even be intentional since it's supposed to be read by the DM and he can decide whatever he wants. This means it's probably implied that increasing the maximum score by 2 (to 22) allows you to increase your stat score to 22 by any mean, including ASI.
A further indication that you should be able to do it is that every mean of increasing your stats says something on the lines of "up to 20" (AFAIK), instead of "up to your cap". This means any feature increasing your cap doesn't do anything, since everything is hard capped in its own text anyway (Ioun Stone as another example besides ASI).
For clarity, what I mean here is that there would be no reason to word it as "your maximum for that score increases by 2" instead of "up to any value" or "not hard capped at any value" or anything like this if it was not supposed to influence any other item or feature.
About your last question
In fact, is 20 even the maximum for your ability score when using items; the quote seems to only apply to Class Features.
The quote is a general rule. This shouldn't matter though, because as far as I know, the items explicitly state their cap (Deck of Many Things' star card caps at 24, Ioun Stones cap at 20, etc).
Probably not (except for some minor customization).
As you said, many True Polymorph shenanigans fall under DM fiat. This includes whether you can polymorph into a specific creature (and not a generic example of a specific kind of creature), and others of the sort.
If you turn a creature into another kind of creature
You, the caster, can essentially choose a kind of creature. While the DM has leeway to create new monsters for his world and adventure, players (in every table I've ever been, but again, there can be exceptions) don't usually have the power to just invent a new kind of monster.
I would adjudicate, for your server, that you can only True Polymorph a target into one of three:
an officially existing creature, taken from the MM, VGM, MToF, etc. This includes variant options of monsters (like the Shadow Dragon). All of these should still have the approval of the DM, who may claim a given creature does not exist in his setting
an officially existing NPC, with a specific (DM approved) race. Appendix B of the MM and of VGM describe class-based NPCs (Bards, Warlocks, etc) as any race, and claims the DM can can add racial traits to further customize them. it seems logical to assume that, when True Polymorphing into one, the DM (or player) could logically pick an adequate race.
a homebrewed creature that the DM has created and that the players know exists in the setting
This basically prevents players from being able to make up some creature that fits exactly all the requirements they need for some specific problem, it gives the DM a basis for obtaining statistics for creatures (either from the official sources or from their own notes), and it also gives the players a fixed set of sources (the books and the DM's notes) from which they can draw ideas and kinds of monsters.
Being a server, this should be a proposition that follows your social contract (e.g., all DMs agree on it, or something similar).
Best Answer
Yes, there's a way to increase ability scores with a class ability
At 20th level, a Barbarian gains +4 to strength and +4 to constitution as their capstone ability. This meets your requirement of increasing ability scores without items, but is going to prove unlikely to help with increasing a Warlocks damage from lifedrinker.
There is the option of casting the 9th level spell Shapechange, which allows you to change your strength, dexterity and constitution, but once again, this won't help you with your specific problem, because you keep your own charisma. It's also a 9th level Wizard / Druid spell, so it seems unlikely you'll have access to it as a Warlock.
For that one, you're most likely out of luck. D&D 5E has done away with almost all effects that increase or decrease ability scores, to streamline the game and to make it so you won't have to recalculate all your skill and attack bonuses on the fly.
There are still a few monsters that can drain ability scores (the Intellect Devourer and Shadow come to mind), but as far as I can find, there are no official spells or abilities that increase ability scores asides from the Barbarian capstone. Those kinds of things have all been replaced with the advantage mechanic.
PS:
On a pure technicality level, Ability Score Increases you get at certain levels are also technically a class feature, so those are also an option, as pointed out by @DaleM, but I assumed you were already aware of those.