SOMEBODY HELP ME!! need some advice on how to deal with my ex-girlfriend's blatant audacity. 2 yr old involved

My girlfriend and me were having some problems and she broke up with me. It is already a difficult breakup because we live together and have a two year old son that she wants custody of. I want joint physical custody because I believe that both parents in his life as much as possible is what is best for him. The other night I got a big bill from from my phn company so after doing some investigating come to find out that from that she had made calls and text messages to her ex boyfriend for over 6 hrs 3 days prior to this. When I confronted her on this at first she denied it but I had it all in black in white that her calls were while I was at work or sleeping. I accused her of cheating on me and said if we were going to make it in this house together until this is over she was going to have to stop lying to me because she had broken her promise to me to not talk to other guys or ex-boyfriend until she had moved out. A promise that she obviously was not able to keep