As James says, a visa can't be transferred. Carry both until it is time to get a new one, or if an extension of stay is applied for, that can be stamped in the new passport later.
Why is it belittling to follow a process to get a visa to have the privilege of staying in Thailand? I hope they don't ask for tea money when they visit. Not sure if that happens much. Good luck.
You may need the mother, but you DO have legal parental rights. You should be able to go with your child and apply by yourself and all of the documents, blue book for the kid, etc. Check at your local IO, as you never really know what they will want when it comes to kids.