I’m afraid that you and your long distance girlfriend have hit a deal breaker in your relationship that signals the end of it. It would be a mistake for you to move to Corpus Christi for her knowing that your career would be severely hurt by doing so, and your not liking that city as much as where you are now, to boot. She’s made it clear that she doesn’t want to move to Melbourne where you live because of limited job prospects there for her, so you’re both being very clear that this relationship that survived long distance for three years, is now done. 🙁
What you aren’t saying, but what’s coming through loud and clear is that both of you have made your careers your priorities, and your relationship distance fit your careers. There’s nothing wrong with that, but now she’s wanting you to move unsettling the balance and the status of the relationship that’s worked for three years.
I think it’s going to be better for you (both) in the long run to let this relationship go, and next go round, look for someone local to date. Geographical compatibility is important.