If I were you, I'd just take my bike to another shop and see what they say. If it's really just a hub adjustment, it's a quick and cheap fix. Probably around $25-$30. It's a 10 minute job. Shops are busy so they may not be able to do it while you wait, but I would expect no more than a few days turn-around on something like that.
If it's an easy fix and another shop can do it, then your problem is solved and the only remaining issue is the money. If I was in this situation I'd pay the second shop, then go back to the first shop with documentation and ask them to refund the difference.