Unmarried- baby’s last name

I have a question for unmarried mama’s that have had babies already:

What was your experience with your baby having/not having the fathers last name?

Yesterday my doctor told me that because we’re not married yet, our son will be born with my last name even if the dad is there & signs the birth certificate. We’re due to get married in June but don’t want to go through the process of changing the baby’s name from mine to his.

We’re now contemplating getting married within the next few weeks to start the process of changing my name, the issue is that I’m due in 4 weeks at most😅

Any mamas had similar experiences or want to give some advice? I’d really appreciate it😊