Dad Died In Georgia With No Will. Am I Next Of Kin If My Parents Never Married And/or Can I Prove He’s My Biological Father?

As title says, my dad suddenly passed in and lived in Georgia while we are in New Hampshire. I did not know if he had a will, but assumed if he didn’t, his assets would go to me and my brother (both adult children) since his wife had previously passed and we would be next of kin. His friend who he lived with, “John” for the sake of the story, has not been too cooperative and has given conflicting information about what’s going on, but says there was no will that he could find. My mom was contacted by John’s probate lawyer saying my brother and I are technically “illegitimate children” since my parents were never married and cannot be next of kin if my dad never legitimized us.

Him not being my dad legally was never something that crossed our minds. He is on our birth certificates which I thought would be enough. How do I prove a deceased person is my biological father?