From googling, it looks like you don't have to change your state of residency due to the military move, so you just put in your current address to vote. Are you registered where you currently reside?
I would assume that the base you move to would help you out with things like this.
Re: alright, military moms (NBR)
I am not a military wife or in the military, but I would just vote absentee:
https://www.fvap.gov/resources/media/fpca.pdf
From googling, it looks like you don't have to change your state of residency due to the military move, so you just put in your current address to vote. Are you registered where you currently reside?
I would assume that the base you move to would help you out with things like this.