You should be able to change your last name without much problem, depending on if you have a criminal record and you are not trying to change your name to avioid creditors.The procedures for changing a last name in Florida require several forms and a court hearing. Individuals seeking a name change must be a legal resident of Florida and must submit to a background check. Last names can be changed for a variety of reasons, but frivolous name changes are not allowed in Florida courts.
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You need to file a Petition for Name Change. It is a pretty straightforward process, and as long as you are not changing your name for some ilicit purpose (which it doesn't sound like you are), then the Name Change should be granted. Tehcnically though, even with the name change, your biological father would still not be your "legal" father because of the adoption.
I believe that your question was more about removing a name then about changing yours. I don't believe that you can remove the name from the birth certificate because this is the record of your birth. While he may no longer have involvement with you he is your father from the birth certificate sense. When an adoption occurs it is as if the adoptive parents are the birth parents and any previous records are destroyed. You could ask to be adopted as an adult by your biological father.