He sounds a little possessive and controlling. Not good traits. If he is not respecting your feelings about PDA and your time, he probably isn't the guy for you. Two months is such a short time for you to be having such doubts. Chances are there is a better match out there for you. I would recommend asking him to respect more of your personal boundaries and realize that you have friendships outside of the relationship that you wish to maintain and if he can't handle that, he isn't the guy for you.