I don't know my MMOs well, but most of them have an RP scene. But it's largely player-lead.
If you want to get into *roleplaying* rather than *rollplaying* for real though, the best advice I can give is to get involved with a good Tabletop group. Adventurer's League and the like are pretty rubbish for this usually, but if you can find a party or get some friends together or something you can get that "playing a character" experience far better than any computer RPG.
But in the realm of singleplayer gaming, there are still computer games that let you embody a character and get invested in a story. Bioware games tend to do this well—it's kind of what they do. But seeing as you're playing their MMO, you probably don't need me to tell you that, nor do you likely need me to tell you to play KotOR, Mass Effect, Baldur's Gate, or Dragon Age. But I'm telling you anyways just in case.
The great Western RPGs are kind of all built around this to a greater or lesser degree, but early on most of them starred cardboard PCs that were little more than standins for you. I mean, unless you're playing Ultima, and in that case, the player character literally is you. Which was cool in Ultima IV, but afterwards largely just happened because Ultima IV did it and it didn't serve to provide any substantive message about being a moral person or anything like it did in that original game (don't... mention Ultimas 1-3. I mean, you can, there just isn't much point). Prior to the mid-to-late 90s, roleplaying was really the province of weirdo experimental text adventures, and if you're not an oddball like me you'll think those have aged pretty badly. Visual Novels also had some kinda-roleplaying-ey stuff in them (Fate is very good at forcing you to get into Shirou's mindset or die, at the very least...) but are in no way what you're after and really the medium constrains them from doing too much there. So I'm going to take this from the top.
If you want to play RPGs that let you play a character, really roleplay, you have to start with the aformentioned Bioware games and the games of Bioware's cousin, Black Isle Studios. Who created Fallout 1 and 2, and the much-celebrated Planescape: Torment, which is probably the closest to what you want: sometimes dubbed "the best book you'll ever play" it's full of words, words, words and words, all spoken by exceedingly memorable characters. Like yout talking skull companion who I swear must have been an inspiration on Dresden's Bob The Skull.
These games aren't always the best at roleplaying (they have a strong combat focus, etc) but they do give you the chance to embody a character and inhabit a world that feels pretty alive.
Black Isle died in the early 2000s, but its spirit is kept alive by its two child studios, Inxile (the torchbearer for That Oldschool Fallout Spirit by virtue of being the team behind Wasteland 2, and the people behind Torment: Tides of Numenera, the in-all-but-name sequel of Planescape) and Obsidian Entertainment (the team who actually got to make Fallout: New Vegas and put out Pillars of Eternity, which is very much in the same densely-worldbuilt kind of vein of Torment and so on). Obsidian's offerings are probably going yo be of more interest to you.
Although speaking of worlds that feel alive, The Witcher games are pretty amazing at delivering that. Even if Geralt is... less than the most interesting to play (although he is very attractive, as Iximaz will attest).
A lot of people point to Bethesda games as great games to roleplay in, and... no? See, I'm of the opinion that Bethesda games are more hack-and-slashy games than RPGs per se (albeit with an extremely well-constructed lore around them), and I've never felt like anything other than a psychotic murderhobo while playing one. It's telling that of the 3D Fallouts, it's the one made by Obsidian that is generally most beloved by the fans (which makes sense: they're staffed with people who literally built that franchise. And then Bethesda didn't give them a bonus because of Metacritic...). What I will say is that you totally can roleplay in a Bethesda game. I know plenty of people do. But I don't feel like the game will help you along.
No, I don't hate Bethesda, I just feel the need to balance out the undeserved unconditional praise they get.
I don't think I've really missed any big name in RPGs...
Except. There is another kind of roleplaying: Experiential roleplaying. If you want to play a game and rather than customizing, building, or embodying a character more really just... feel like you're really there, there is one company that could do that better than anyone else and its name was Looking Glass Studios. Thief: The Dark Project (and it's re-release Thief Gold) and its sequel Thief 2 are stealth games, not RPGs. They are build on an old, creaky, graphically outdated game engine from the late 90s, the look terrible, and in some cases you have to download a fan patch just to make them run. However, if you really want to feel like a sneaky thief breaking into the houses of the wealthy and powerful in a weird fantasy city, I can't recommend these games enough. They are simply incredible pieces of game design, and they really make you feel like you're there.
While Looking Glass itself really only made RPGs on DOS that are so old and ugly I can't expect anyone but me to play them, they did parter with a child studio of theirs, Irrational Games (yes, that Irrational Games) to make System Shock 2, widely regarded as one of the greatest RPGs ever made. And while it's not the best for roleplaying (characters don't exist because everyone is dead except you), it's another game that really creates that "like you're really there" feeling, in this case of oppressive dread over being locked on a spaceship with a psychotic evil computer who hates you.
Irrational went on to make the Bioshock games in the same vein, and everyone loved those, so... yeah.
The other Looking Glass alum RPG worth mentioning is Deus Ex, the yin to SS2's yang and the game that's usually mentioned in the same breath. Lots more opportunities to roleplay here, even though it's primarily an action game. You may not be interested in it. I dunno. My advice? At least try to play it anyways. The Steam Summer Sale is starting soon, and Deus Ex regularly goes for 99¢. So does SS2, but that game is harder to recommend because it's really hard to beat and doesn't control quite as modern.
Actually, most of the games I've mentioned are old and thus cheap during summer sales. Most of them are worth it. Hopefully, some of them scratch your itch. The BioWare ones probably will, Tabletop games definitely will, the rest... I am less sure of. It really depends on what you want out of a roleplaying experience. But with luck, one of them will.