Look, i am a bit bias in this situation considering that i got lied to and cheated on recently. I think trust is very important when u care and love someone and are serious enough to be moving in with them...but lets face facts...people sometimes lie and people are full of surprises. You might think you know this guy very well, but he might be lying about a few things behind ur back. I'm not saying that he is indeed cheating on u or lying, but i just think u should take extra precautions. If u find strong evidence that something is going on, then stand by ur instincts. I totally trusted my ex despite all the rumors i heard about him cheating on me. I had never trusted anyone and he made me feel bad about questioning him so much. Until one day I finally told him "look, people talk rumors, but rumors don't sprout from the ground...especially if the whole world is talking about it.!" and he denied and denied until he saw no other choice but to say the truth. and even after that, i kept hearing new things everyday. and he slowly admitted to them. I decided I'm not going to live my life wondering what he really does behind my bak and interrogating him forever. i just let him go.
Well, just be careful, don't interrogate too much. I think its okay to do some of ur own "research" (no spy, but just find little details out).
*GOod lUcK*