I live in Texas, and my partner & I agree to be married under common law (present each other as husb/wife), can he claim me as a dependent if I did not work part of 2018?

You cannot have it both ways. A spouse is never claimed as a dependent. Either file a joint return or both file as single. 

So, even if we are not legally married, we can still file jointly? Would it be more beneficial as  I made about $14K?

If you made over $4150 you cannot be claimed as anyone's dependent.  I do not know the specific requirements in TX to claim to be common-law, but joint is almost always a better way to file.