Can you find a real relationship online? Am i being paranoid?

i got chatting to this guy on a site and after a few months we met up, i had a few issues getting with him like he is very skinny which is attractive to some but not my type. Anyway not being a person caught up on looks, my biggest issue with him is he drank vodka nightly to ease his pain (he has spinal bifida) We clicked really well personality Wise and eventually started seriously dating...5 months later and i'm living with him and really believe i love him and he hasn't touched any alcohol as he knows it upsets me....but....i found out he is still chatting online and telling girls he is single and lives alone....and he was txting y messages to one of the girls he gave his number to. I confronted him and he said it was just something he does to p the time, ans as for the y texts...guys just do that and it doesn't mean anything. I still get upset and yes, jealous, always worrying who he is texting. I hate not trusting him....AM I BEING PARANOID?? is he right that what he did is ok, or can i call that cheating?