This is a common problem, and quite a difficult one to tackle. The difficulty now is that both of you will be tense around bath-time, and your daughter will quite possibly pick up on your anxiety. I think it would be advisable to not even attempt bathing for a week, and just wash her on the changing mat as you have been doing, or top and tail. Then try again after a break, but be as relaxed as possible. It would be a good idea to try swimming and be very encouraging and confident.
Babies are quick to pick up parental fears and feelings.
All the best with this. As regards hair washing, this just needs to be done, and a plastic face guard could be purchased to avoid splashes on the face.