Well, the fuel sending unit is bad in my '83 (imagine that).
I know they're available for the later fuel injected cars with electric fuel pump, but what about my '83 with carb? Am I just screwed?...

 

I would try looking at Dorman, Standard, OER, 4 Seasons, or Blue Streak.

If you have no luck, you could switch over to using a 4th-Gen Tank and Sending-Unit.

Otherwise you could go to a Performance-Aftermarket solution...
such as a Sending-Unit (with or without a Lift-Pump) from Holley.

I have been switching all of my Customer's 3rd-Gens to the 4th-Gen Fuel Tank...
So I have not purchased anything for a 3rd-Gen Fuel Tank in a very long time (Plus these old metal Tanks are usually in poor shape/ rusted and nasty).
Most of my Customers go with a Holley System... if not, then I use a Stock ACDelco Sending-Unit.

 

Forgive my ignorance, but don't 4th gens also have the electric fuel pump in the tank?...

 

Yes, you would have to regulate the Fuel-Pressure down if using a Stock 4th-Gen Sending-Unit...
and remove the Mechanical Fuel-Pump (also install a Block-Off Plate).

The Holley Sending-Units come 3-Ways...
-No Pump.
-Low Pressure Pump for a Carb.
-High Pressure Pump for EFI.

 

So the Holley sending unit would only fit a 4th-gen tank?...

 

GM 25003095 Sender, genuine OEM part (oemcats.com) Maybe?

 

Thanks, but unfortunately that is for the crossfire injection models...

 

They also offer Universal-Fit Models...
Fitment with those can be a real pain in the A$$ though.