Facebook and relationship can work? *SOS*

I really really need to talk with someone with a successful relationship to tell me how she deal with her partner's facebook account.

Im new in a serious relationship and he asked me to move in yesterday.. I know that should make me thrilled, but... I am worried that if we don't deal with current problems we will increase the current ones by living together.. As I wrote I am worried about facebook, already found that he was texting a girl he tried to date before me and when I asked him he said his intentions were friendly but he also admitted he did that for his self esteem. I didn't know how to get that so I asked him to prove my worries wrong and he is trying a lot since that day.

But my worries remain. I ve been with men who lied a lot so I can't relax easily my mind anymore.. How are you making your relationships work and using social media? Is your partner having access to your facebook or are you to his? (I'm getting this a lot lately from couples)

Should I ask him about girls he follows on instagram? I don't know how to deal with it maturely.. Any advice will be a real help right now