Active Army: Is it possible to go through any sort of official channel to request a change of Unit?

For the unit swap you will need to find another 15W SPC who wants to PCS to your unit. Unless it changed I believe if that gets approved it’s on your dime to move there.

When it's time to reenlist you can reenlist for a new duty station.

I think that the Army Times also still posts the Stateside Swaps. You may want to look and see if you can find someone who is willing to swap units with you.