My fiancee and I plan to get married next year, if the biological father signs over his rights and I adopt them do they get my last name?

Yes. Contact an attorney for the details of how to proceed.

The court will not allow someone to just sign over their rights. There must be a termination of parents rights, which the court will often only grant if there is a step parent ready to adopt the child. The only other option is a consented adoption, which does the termination and adoption all at the same time. But it requires that you be married. The court will not allow a fiancee to adopt. If you do an adoption, you choose the name the child will have (technically, the adopting parent, being the petitioner, is the one who "chooses" as they are the one bringing the case and signing the paperwork).