Child custody and visitation in Texas is governed by the "best interest" of the child. The Texas Family Code says "the best interest of the child shall always be the primary consideration of the court in determining issues of conservatorship and possession of and access to the child."
Don't buy into the child support threat. I spoke to a young mother today who had a three year old child whose father was not paying child support. She said he wasn't paying child support and I asked why. She told me that he threatened to file for custody and take the child away from her if she "tried to put child support" on him. This fellow had three other children by two other women. He did not have custody of any of the three.
The full division of Family Trial Courts are open and operational today, including uncontested dockets, ancillary dockets and already scheduled trial dockets. Call the court in which your case is set to determine start time.