There's a HUGE difference between sole custody and giving up your parental rights. If one parent has sole legal and physical custody, the other parent may still be entitled to visitation as specified in a court order, is still entitled to access the child's education and health care records, at least in California, and is also still obligated for child support. The "non-custodial" parent can still ask the court to modify the custody order and change it if necessary for the best interests of the child, and if something happens to the custodial parent, the non-custodial parent is still the legally preferred next in line.
If a parent has relinquished all parental rights, which usually happens in the context of an adoption, and NEVER happens without some sort of court intervention, then that parent's parental rights and responsibilities end completely from there forward.
If you do not want your child to be taken out of state, then you do not want to give the custodial spouse full physical custody. Having full physical custody gives the custodial spouse the presumptive right to relocate out of state. You need to defend your interest in the child if you fear that you may be denied access to your child later. Moreover, it depends on what type of criminal record you have. If it is for domestic violence, molestation, or something along those lines, then yea, it could affect you. I would look into getting an order from the court outlining your custodial rights and arrangement.
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