Is he cheating - Help?

I've been dating my boyfriend for almost three months; we met on a dating app. We spent the holidays together on a road trip, which was amazing, and things have been good. We are supposedly exclusive, spend almost every night together, and our kids get along. We both have some baggage from past relationships, but so far we have been able to work through things and haven't had any major problems.
Last night I got an email from a woman who said that she has been communicating with him on another dating app, and that she just found out about me from a mutual facebook friend. She sent me a screenshot of their conversation and said that once she confronted him about having a girlfriend, he stopped responding and then his profile disappeared. Sha also said she has seen his profile on two other sites. The conversation is just small talk; nothing sexual or even trying to meet in person.
My boyfriend says that he is being framed, that the profiles aren't his, and that he has not been talking to any other women (the conversation is fake). He completely denies all of it. The pictures used in the profiles are on his facebook page and/or instagram, but both of those are set to "friends only" or "private". But all of the information is spot-on him, including that he doesn't drink which just happened recently.
Should I believe him? I don't want to keep seeing him if he is still shopping around, and that wouldn't be good for my kids either.
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Today I got an anonymous letter in the post with my address listed as the recipient and the return address. It said my boyfriend has a history of treating women poorly, and looking for new girlfriends before he breaks up with the old one. It also said he was online talking to women. Finally, it referred to my recent divorce how hard it was for my kids and I (whoever sent it knows something about that situation). The writer said they didn't want telling me to negatively impact our relationship.
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Second update. Today I got a facebook message with screenshots of a conversation he supposedly had with a woman (the woman he says is framing him) where he set up a lunch date. We only see each other in the evenings.
Is he cheating - Help?
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