GRS is just a surgery, but outside of sex, it does not change ones life if one is already living full time.
I really do not see the need to move away and start over, but considering you live in the south, it is best you move to a state where you actually have legal protections and are not considered a second class citizen, with or without surgery. As stealth as you may want to be or think you are, it matters very little if one does not have rights and is surrounded by the conservatism (and often bigotry) that is so pervasive in the south. My advice, is move on, but move on to a place where you have legal protections. I would look to the states that have passed gender identity laws, not single one of those states is southern.
If you want to move on realize there is paper trail regarding your past, whether you realize it or not. One can only be so "stealth". This is especially true in the south, where transsexuals are treated as second class citizens or worst.
My advice...West Coast. MN, IL, Iowa, New Mexico, Colorado, DC, VT, NJ
There you have rights. In the south you do not. Plain and simple. Move north, move west, or move east, but get the HELL out of the south. No transsexual, or queer person in general, should live in the south. It is to live without rights. It is to live under bigotry, queerphobia and conservatism. If you have to move...get yourself out of the south in general.
PS I live in DC. I have rights here, including the right to change my ID without surgery. But also numerous protections that go beyond that. If one has to live pre-op, or even post-op, it is best to live somewhere where one has full protections under the law. The south will probably be the last states to enshrine those protections, in fact it will probably happen on a federal level before a single southern state passes legal protections. As such, I think no one should live or work in the south if they are transsexual. You are wasting your time, and exposing yourself to legal discrimination.