How do I legally change my child's last name without an attorney ?
Can I legally change my child's last name on my own by making the proper forms and submitting them to the courthouse ?
If my child was born in Florida and we now reside in Texas, do I submit the forms at the courthouse in Florida or Texas ?

First things first: is there a father listed on the birth certificate? Is there any type of custodial agreement in place which may allow him to have a say in the name change?

No, my ex husband doesn't brother to have any contact with our son at all.

Just because he doesn't have contact doesn't mean that he has no rights. What does your divorce agreement say about custody?

If his name is on t he birth certificate you will need to get his permission or show a good faith effort to if you want the court to allow the change.

Go to your local Family Court and ask what the process is.

He didn't appear for court for the divorce. I have sole custody of our son and his father didn't sign the birth certificate.

his father didn't sign the birth certificate.

Doesn't matter whether he signed it or not. If he is listed as the father, he has give permission. If you think he will not show up in court, then serve him with a summons notifying him that you are petitioning for the name change. If he does not show up to protest the judge will grant your motion by default.