Balmung, Mateus and Omega are the most prominent but there's pockets of role-players to be found on other servers such as Cerberus if you do a bit of digging. Unfortunately there are a lot of flaky individuals in FFXIV's role-playing community so even on the busier servers it might take some time to settle and establish some reliable contacts.

There's also a site devoted to RP in FFXIV, as found here:

https://ffxiv-roleplayers.com/

It is, admittedly, more geared towards Balmung and Mateus...though players from other servers do pop up from time to time.