Walk-ups are generally not a thing, sadly. I’ve only managed to catch people RPing once, two elves and an orc in Orgrimmar! I tried to join but was abruptly dismissed, somewhat rudely. I think people here just aren’t used to it. Your best bet is to join a guild if you want RP. Generally speaking, the quality is much higher than on servers like AD and such (in my very subjective personal experience, naturally), but people are also a lot more invested and will expect the same of you. Unless you join a casual guild, of course!
If you too wish to flee to SGC, I would refer to this thread for active guilds and see if any description suits your taste. It’s up to date as far as I know: [RP] The (New) Roleplay Guild List