Please seek a consultation with an adoption attorney. The first issue is whether your current husband wants to become legally responsible for your daughter. The second is whether the father is in agreement with terminating his rights. If this proceeds you will need to determine if the name change is a requirement of the proceeding or what you can do to keep her current name.
Loretta is correct.
The answer to your question is yes the name can be changed, or not changed. But there are a lot of issues you face - such as support, custody, visitation. A lawyer can help advise you about your options and rights, even if you do not hire a lawyer. At least you know.
This is general legal information, not intended to apply to your specific case. And I may not be licensed to practice in your particular state. Under Federal Law, I am a debt relief agent.