[MomKP]

I had a son with a man I was seeing for about a year. I told him when we met that I was married but we haven't been able to afford a divorce yet and that him and I remain good friends. He lived in a different state and everything.
Time goes on and I fell pregnant. When I was pregnant I asked him for our son to have both of our last names. He refused and said absolutely not. We were not married and I didn't know if we were going to work out. After much arguing we gave our son my last name. However, we didn't know that he could not be on the birth certificate because I was legally married to someone else.
We broke up 3 months ago, come to find out he is remarried and moved on. I just got served papers that he what's 'timing-sharing' which I was doing until he stop supporting in any way shape or form. He also want his last name change to his (the child is 1 now). And what's to be on the birth cert. which I agree to that too.
My questions are: can he have his last name changed? Can he be on the birth cert. if I am legal married to someone else?