I would put the brakes on moving in together until the trust is re-established.
Do you really want to live with someone that you don't trust who was actively looking for people to cheat on you with???
If he loves you then he'll be willing to do whatever needs to be done to be faithful and build the trust back up. That includes counselling, stopping flirting and making out with other people, and taking the time to do those things before he's living in your space.