Should I tell my research advisor I go to therapy?

I would set up a personal meeting with her so that she knows to take the situation seriously. If you send an email or say it in passing she will easily forget about it. And tell her the absolute truth. If she does not appreciate the honesty that youve brought to the table or if she begins to treat you differently or unfairly you should be able to go above her and talk to her boss or whomever. Its better to let her know early than it is late and let something build up.