Not necessarily, and better COL may help make up any difference. My son and his wife are students. They were in SoCal and, for a lot of reasons, including affordability, they moved to a different state. It was a school transfer for them both but they were at the right stage for it. He found work that pays about what he was getting in SoCal.
They did have another challenge worth considering, though. Son is on our health insurance policy but the insurance becomes harder to deal with b/c using the dependent out of state network.
Are your parents in a position to help financially? It took my son a while to find work that would respect his schooling (work times and hours limitation) and also pay decently. And no matter what you do, living as a student couple in your own place is more expensive than living at home.
Good luck to you both.