Charged with abandonment- for not paying the full amount of child support- due to job and
Income change. Divorce took place in Georgia and I had to move back to Texas where my Family lives. I did not have a lawyer and she did. They also file a contempt of court on me.
I need to go back to the court and file for a modification of the custody and support, but if I go back to Georgia I was told I will get picked up, but Georgia Law we not come to Texas. What do I do and how should I go about getting this taken care of.

Try to get the matter transferred to where you are - but I don't think that will happen.

The time to file for a reduction (at which time you would show a change of circumstances) was before you moved. You ARE in contempt and, as such, yes, you can and probably will be arrested.

Go back and try to negotiate something - you will owe the "old" support amount until the Court orders a change. Every day you delay the arrearages pile up a little bit higher.