Which next base should I seek? Also, what are the E-7 board members looking for when they select someone for E-7?

I can vouch for Alaska (JBER) being a great duty station, at least as far as location is concerned, didn't have a great experience with 4/25. I absolutely loved my time in Alaska, and the summers are to die for. I can vouch for 10th MTN at Drum, loved my time in 2-22, lots of good, hard training. Drum itself is kind of isolated, but you do have everything you would want from a city in Watertown, and Syracuse is 90 min away and Canada is 30 min away with the 1000 islands just short of that.

I agree with seeking a broadening assignment. Recruiting or Instructor. Ensure you are filling up your military education, completing 40 hour course every two years with a focus on those courses that are valuable to your career field. You should also be working on completing an AA, this will place you above your peers. I see the only real leadership positions you’ve held are Team Leader and Squad Leader? As a SSG you should be seeking a section Sergeant position and hold it for two year while getting strong NCOERs. If you want to make SFC, you’ll need Senior Rater comments that say you should be promoted ahead of your peers. Your NCOERs will have to be complimented by challenging positions as well. Good luck, it seems like you have some time since you just pinned SSG less than a year ago.

You know, if you're reenlisting indef, there's no choice of duty station. You will have to work it out with your branch manager.
If you haven't yet, you should be looking at a broadening assignment as opposed to duty stations.