I use to live in Fl and now I live in TN. I'm giving birth to twins in December. The father is willing to give up his parental rights (this is not about child support).
Do I need a lawyer to file the paperwork or can I do it myself?
If down the line I get married, can my husband adopt the kids or do we need to seek the father again even if he gave up his parental right?
Also, if I give the twins my last name at birth and do not name a father, how would that work? Does the father have any rights then?

First. The father can't just give up rights. This has to be approved by a court of law. And family courts are very reluctant to grant a Termination of Parental Rights (TPR). Generally they will only do so to clear the way for adoption or if the parent represents a danger to the children.

There are hundreds of threads in this forum that all say the same thing.

Now, if you don't name the father, then he will have no rights unless he challenges and goes to court for them.

However, if you ever need to apply for publis assistance, they will insiste you name the father andf they will go after him for support.

If I do not name a father in the birth certificate and I marry, can I name a father afterwards, even if he is not the biological father?

If I do not name a father in the birth certificate and I marry, can I name a father afterwards, even if he is not the biological father?

No.

That could lead to arrest and conviction for providing false information.

A birth certificate is an official document. Lying on an official document is perjury.