You said it yourself, you don’t trust him. And I think you have a very good reason not to, he’s cheated in every relationship prior to you, so yeah, the chances are pretty good he would cheat on you too. And you also know that words are just words, and guys go on and on about marriage and kids and houses and then they cheat.
You’re asking how to get over your lack of trust but that just isn’t a thing. It also sounds like if he never went out without you, you’d be fine, but that’s not a thing either. If you always had trust issues because of past trauma, that’s something you could work on in therapy, but it sounds like you’re in a relationship with a serial cheater, who you don’t trust not to cheat on you. I suppose you could do some soul searching about what he could do to earn your trust and then communicate that to him. But if it comes down to “please don’t go out without me,” then that’s not reasonable.
Definitely don’t move in with someone you don’t trust.