AAMCO might be able to rebuild a Turbo 350 automatic But I question most of there shops abilities.

I have been in these boxes as well as other ones a few times over the years.
It is risky replacing one half of a failed gearset.
There is always damage to the mating gear. If the tooth form on the non replaced gear visually is fine then it should be magnafluxed to insure that it is not cracked at the base of it's teeth. Did they do that, I didn't think so.
If you reuse a gear from a damaged set I stone all imperfections in the teeth. Material from the broken gear can get transferred to the other one. It needs to be removed. Did they do that, nope, otherwise they would not say it will be noisy for awhile.
The broken bits will many times pass through some other gears, they also must be checked for nicks. If they show a nick they must get stoned smooth also.
That noise is the non replaced gear chewing up the new one.
Bearings, were they ALL replaced? An aftermarket bearing kit is about $140. Little flecks of metal can get in any bearing as well as the needle bearings under each gearset. Being Scottish I made that mistake. I replace them all now.
It takes a few hours to save a gearset, it is easy to cut corners but that can rear its ugly head as yours has.

Since time is an issue you may want to look for a used box, drop the bottom cover at the minimum and look in it.
You can check the syncros with a look and see the gearteeth but can learn little if a bearing is noisy.
These are not a difficult gearbox to work on, there is some presswork to get it fully apart and together.

Hopfully this info will be useful when you go back to "discuss the issue" with the shop.