The previous owner installed 3.73s in my 93 Firehawk.
The whine is really getting on my nerves.

Assuming these have probably been in there for a few thousand miles;
Is there any way to fix that other than paying a shop to replace
the gears ($600-1000)?

 

Improperly installed gears must be replaced if they've been driven for more than just test purposes. Once a gear set with the wrong pinion depth or backlash has been broken in it'll whine for life, regardless of whether or not someone fixed it later.

 

That's pretty much what I figured. I'll deal with it for now.
And hopefully the gear will die soon and the extended warranty will fix it.
(covers mods in place at the time of purchase gotta love sqaretrade)