Thanks for your service! That being said, you are mandated by congress to attend some form of TAP class, whether or not they want/allow you to go to the weeklong/days long class or not is something separate . At one of these you have to get DD-2648 form, which you need to get your DD-214, it is required by congress, and will be asked for at PSD before you get out. It actually says on the form: "Title 10, USC 1142 requires that not later than 90 days before separation, preseparation counseling for service members be made available." Required! Too many vets got out and didn't realize all the services and benefits available to them through VA, Department of Labor, etc. Please talk to your Fleet and family support center/Navy counselor asap to fix that.
I was looking at NRC, but I know at least one opportunity had a minimum of a Bachelor's degree from an Engineering accredited school, (basically saying no Thomas Edison style degree is what that specific job posting was saying) so hopefully you have a 4 year degree from University of South Carolina or a similarly accredited good school per NRC standards. I'm researching that now, with my wife's uncle being the former deputy director of the NRC. (weird)
As a general rule, areas around bases have fewer job openings because other people want to live there, too. Therefore, if you move, it is less competitive with 100 other sailors trying to get the same job, should be easier (my opinion).