Understand that we cannot give legal advice on here, other than giving you general information about the law. In answer to your question of whether dad can try to get full custody of your child, he can try. The question is will he succeed?
In order to change custody, the non-custodial parent must prove two things. First, he must prove there has been a material change in the child's circumstances since the last time a custody order was entered. For example, if the custodial parent is now on drugs, or is living with a boyfriend, or has abused the child, or something else that affects THE CHILD's circumstances, then that is the first hurdle.
Once the non-custodial parent proves a material change in the child's circumstances, the second step is for the judge to then decide whether it is in the best interests of the child (not the parents) for custody to change. In order to decide material change, many things go into that, including whether the child wants to move households. The older the child, the more likely a court is to listen to their preferences, if noncustodial dad is stable. But child's preference is just one factor for the court to consider.
If he has filed a motion to change custody, you need to hire a good family law lawyer who will file an answer on your behalf. Also, if custody does change, when the noncustodial parent owes back child support, the court will generally give the new noncustodial parent a credit on child support (yes, you would most likely have to pay child support) up to the amount dad is in arrears. In other words, if dad owes mom $5,000 in child support and mom's child support after a custody change is $100/wk that she owes to dad, she would probably be given credit for the first 50 weeks until that $5000 is setoff.
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