Does child get father's last name if parents are unmarried?

Long story short, my ex fiance and I coparent 2 kids together. We were previously engaged but did not make it to marriage due to cheating on his end. When our kids were born, I put my last name on their birth certificates and planned on changing it to his last name once him and I got married but now that we have broken up, he is still wanting their last names changed. We have a civil relationship. What is the right thing to do?

Does child get father's last name if parents are unmarried?
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