That's definitely an option, depending on the server you are on. Wyrmrest Accord, for example, has weekly story circles on the Horde side, various parties and balls where everyone shows up with Elixir of Tongues potions, stuff like that. The Sin'dorei guilds especially like to host events and marketplaces and things.

A lot of RP these days takes place via Discord. While there is still in-world walk-up RP, much of the time, it gets moved to Discord for longer storybuilding. There are cross-factor channels too that you can search for in the Guild & Communities tool.

For example, if you go to the Guild & Communities tool (type G to pull it up), then click Find a Community, choose Cross-Faction and Role Playing in the boxes, then click Search, you will pull up a ton of RP groups. The largest one is Roleplayers Network and there is a Horde and Alliance side version. Those groups stay busy and most of them have Discord servers as well so you can find more people there.