It is my home, my name on mortgage, ALL utilities and etc are in my name, he pays about $500/mo to "help" that is all. No formal written agreement between us. I want him out as soon as possible.

Since he has paid monthly to "help" he is considered a tenant under NM law and you have to give him 30 days written notice that his tenancy is terminated and he must move out. If he doesn't move out within that time then you can file a lawsuit for eviction.

Yes you do, and if he does not move out within that time frame, you take him to court, it can take 6 to 8 weeks to phsycially get him out if he will not move when told. 4 week notice and then time to get into court and time after court for marshalls to come out and move him.

You of course have no obligation to share you bed with him,