What to do if my exboyfriend claims he’s the father of my child?

If your fiancé’s name is on the birth certificate or he has already completed an affidavit of paternity, then your fiancé would be considered the "presumed father" of your child.  However, this presumption is rebuttable, which means that he could potentially file a custody suit (called a Suit Affecting Parent Child Relationship in Texas) and seek to be legally declared dad and be appointed as a managing or possessory conservator.  If he files suit, he will have to pay the expenses associated with the filing of the case and the expenses of DNA testing.  At this point, the burden is on him-- not you to do anything.

In the event that he does file the custody suit and is the father, you can be working on things now to show that you are the better parent for the child.  It's a good start that you are not an unfit parent—so it's not likely that he can take the child away from you.  At the worse case scenario-- he'll be granted visitation with the child.  However, there are some things you can do and gather info that will help the judge make a decision in your favor.  Show the judge that you are a good parent; that you have thought about adequate child care, good schools, and educational activities for your child.  Have pictures to show the judge of the accommodations available in your home for the child, namely his own bed, adequate closet space (even if shared), and most importantly, everything is clean.   Since you are engaged, it would also help if you and your fiancé could get married before the custody suit to show that you are providing a traditional family unit for your child.  This is not required, but keep in mind that many judges in Texas are still extremely conservative and will frown on unmarried living arrangements.  If it's love-- take the leap and show the judge that ya'll are doing this together.  If you can paint this picture, it will be extremely hard for you ex-boyfriend to get anything more than standardized visitations.  It also sounds like he's a bit of bully to be demanding things that he won't cooperate with-- so you may even want to ask the court to make visitations "graduated" rather than standard (Every other weekend), at least until your child is a bit older.