I'm losing the use of my right hand so the surgery is an absolute
I think you have already answered your question. If surgery is an absolute must, then you only option is to tell your employer just that. Go to your manager and tell them:
I need surgery to avoid losing the use of my right hand. I will be
unable to work for about x weeks. How can we best handle that?
The key point is: You do not ask for leave. You just state the fact that you will not be able to work - no matter what your employer decides.
What happens now will depend on your employer. They may decide to fire you right away - in some jurisdictions you have protections against that, though I don't know about Texas. There is the Family And Medical Leave Act (FMLA), but I don't know if it would help you (you could ask on http://law.stackexchange.com/ ).
However, a reasonable employer will probably not fire you right away. You are presumably valuable to them (otherwise why pay you), so they will try to find out if there is a way to keep you. Then you can start discussing options.
- Maybe it will help if you schedule surgery for a quiet time.
- Maybe you can train someone else to take over your duties during your absence.
- Maybe you can transfer to a different project that starts later.
- Or something else entirely...
Talk to your employer, and offer to make concessions where you can. Of course, it may turn out that they cannot possibly grant you the leave you need - then you will have to quit. Even in that case knowing this well in advance will help you to make the transition easier. But you won't know unless you ask.