I gave my children to their father two years ago because I could no longer take care of them or provide a safe environment for them to live in. Have I relinguished my custody rights?
He is going to request full custody based on the fact that he has taken care of them for two years and continued to pay me child support.

Unless you signed paperwork the children are "simply" living with their father. It is very difficult, in fact, to relinquish custody. The father will probably argue that you were unable to care for them and what's changed over the past two years? He continued to pay you child support even though he had the children living with him and you continued to accept the support? That will not look good for you.

You will go to Court and argue that you can parent the children in a responsible way. He will probably argue that you cannot.

The Court very well may appoint a legal guardian for the chldren, someone who will examne the situation and determine what is in the best interest of the children.

The process varies somewhat by location.

As noted, to relinquish even custody, would require a court order.

If he wants to make it formal that he has primary custody (and end his support obligation) he will need to go to court to do so.

He is just making what you did two years ago, legal. You are still not giving up your rights just giving him custody, and he should have been able to file to stop paying support 2 years ago. Since he should not be paying if he had the kids