Will I get money back for getting charged state taxes from a different state?

You can "tell S1" as much as you want and see how that goes.

To fix this you need to submit a DD Form 2058 to finance, effective date the day you moved to WA.
You get the state taxes back when you file a final return for UT state taxes, DFAS doesn't pay back tax witholdings.

You should contact the tax and revenue office for Utah.

You need to complete a new tax withholding form (W-4) with Washington as your state for tax purposes. To make sure this is entirely legal, suggest you make Washington you home of record with the Army, get a Washington Driver's license, and register to vote in Washington. If you have a car, you should tag it in Washington also. Avoid the three-state problem by at least having your driver's license and car registration from the same state. Transfer the car title also if it isn't too big of a hassle.

Money won't come from the Army, but may come back from the Utah department of revenue. If you changed states during 2019, then complete your Federal tax return first, then the return for the state of which you ARE NOT a resident or were a partial year resident (Utah), and finally the state tax return for the state you live in now (Washington). This is important because you'll be able to deduct the taxes you paid to your former state from the tax bill for the current state. If you moved in 2020, then the two state tax returns will be something you file in 2021.