Three months and he's still seeing this woman? Ok, friendships between the sexes are fine but for her to get weepy etc and mad at him etc....I wouldn't have put up with it this far to be honest.
Ask him straight out and tell him it's causing you disquiet. Whether he will answer honestly or not is another matter and up to you to judge. I think I'd be thinking seriously about getting out. Once you start having doubts, rightly or wrongly about someone's honesty with you I don't think it's ever possible to have a trusting relationship again.