Hello! I need to file a tax amendment, but since my last filing, I have gotten married and legally changed my last name. Will this have an impact on filing an amendment?

Nope ... if you were unmarried that year and had filed a single return and are now amending it simply change your name on the amended return so it matches what is currently on your SS card and the address if it is different and mail in the return. 

Thank you for your help! Do I change my married status on the amendment as well? Or leave it as "single"?

You keep the filing status the same as it was when you filed the original return if you were single for the tax year the return was for.