The name change during the naturalization is for when you want a complete name change (I added a middle name to my name, others use it to americanize their name).
You can change your name to your married name at any point and don't need to wait for naturalization. You can start by going to the SSA office and request a new SSN card in your married name. Bring your marriage certificate as proof of name change. Then change your driver's license and from there you can update your name on bank accounts and other legal things.
Good luck