My question involves name change laws in the State of: Nevada

My husband and I were not yet married, and just barely 20 when we had my son. Perhaps we didn't know enough, or should have asked more questions. But when it came time that the nurse came into my room for the birth certificate information, she asked if we were married. We said no. Then she asked if we wanted to father's name on the birth certificate. We said, absolutely. Then, she informed us that because we weren't married, if we put my fiance's (now husband) name on the birth certificate that it would be a mess for him with the child support office. She said even though we were living in the same house, and he was caring for me, that he would have to fill out child support paper work, and a whole bunch of other things.

Then she said we could always come back after we got married and change his name. So, believing she probably knew better than we did (and maybe she did?) we just went home without my hubby's name on it, and decided to do it later.

We got married roughly 7 months after my son was born. Perhaps because we had gone so long without it being an issue or a thought, we simply had forgotten about the fact that hubby's name wasn't on the certificate. Well, now here he is four, and it was brought to our attention when we put him in preschool. While the preschool doesn't have a problem with the fact we are using my husband's last name for him, his teacher let us know if we want him to use my husband's last name in school starting in kindergarten, we needed to get it changed.

Here is my question. My mother-in-law has told me that if we do not complete this name change on the birth certificate before my son is 5, then my husband, even though he is the biological father, has to adopt him. Is this true? I'm going to get this taken care of as soon as possible, but was wondering if anyone knew the specifics of this before I went in there. My husband is now freaking out, and is feeling very sad that he might have to adopt his own son.

Thank you in advance for your time and advice.