Is it true that active duty 37F will stay at Bragg for most if not all of their careers?

Yes, absolutely true.

Yes. They only have 1 other duty station. There is 1, yes, 1 MOS rep at the Pentagon. Everyone else is stationed at Bragg

No, that is NOT true. While all Active Duty PSYOP units are on Fort Bragg, you will be eligible to fill a planner position at numerous locations around the globe. In this career field, you can dictate your career. If you choose not to take career enhancing/broadening assignments, then you can get stuck at bragg your entire career (and possibly get passed up for senior enlisted selection boards).