These GS500s are aging, and on my bike there was a long-neglected maintenance item that caused a lot of trouble.
I am talking about the pilot mixture needle and associated pieces, this is the "idle mixture screw" many of us adjust to try and get the bike to run correctly at idle and small throttle openings. In my case, problems with it caused a ton of issues.
The picture shows the needle and parts:
(http://www.southbayriders.com/forums/attachments/347826/)
17 - o-ring
18 - washer
19 - spring
20 - needle
The problem is that the o-ring may become degraded, flat, hardened, cracked, etc., just like any o-rings but it may be uncommon to check it. It's very difficult to remove from the carb.
If the o-ring does not seal correctly, then fuel that is supposed to go through the pilot jet will wind up dripping down the needle and out the bottom of the carburetor. This will result in a lean condition. Then if you try and remedy the lean condition by loosening (turning "out") the needle, then it makes the problem worse, not better, since it reduces the pressure applied to the o-ring by the spring and washer, so the leak potentially becomes worse. This manifests as an idle mixture problem that cannot be corrected with the adjustment.
The solution is easy: replace the o-rings. In theory you can do this with the carburetors on the motorcycle, but removing them may be very difficult unless the carburetors are upside down. You first just unscrew the entire needle and take it out. The spring may come out with it. If fuel also drips out at this time, then you definitely had a leaking o-ring. Make sure you retrieve the washer and spring. The o-ring probably will stay stuck in the carburetor. Best trick to get them out is to use a thin piece of copper wire, like about 24ga (from an Ethernet cable?) bent at one end so it makes an "L" shape and pick out the old o-ring. Don't do this unless you have a new one on hand!! Do not lose or misplace the tiny washer!
Then just reassemble the needle, spring, washer and o-ring, in that order, and put them back into the carb. Turn until it stops turning snug (don't tighten hard), then back out 2.5 turns and adjust the idle mixture according to whatever method you prefer.
In one case I forgot to put the little washer in and it leaked badly, the bike wouldn't start. In another case I put one too many o-rings in and it also leaked badly, bike wouldn't start. In another case both o-rings on the carbs were worn plum out and the bike would start and run but it would not idle and the idle mixture could not be adjusted... no matter where you put it, it would run badly.
Given the age of most of the GS500Es out there, I would suggest replacing these o-rings as a maintenance item if you have the carbs off for any other reason. While you're at it, replace all of the other o-rings in the carbs, since they may all be bad. And while the carbs are off, replace the intake boot o-rings since they are also probably shot. I really think the rubber parts in the GS500s have <10 year lifespan, even less if they have been in contact with ethanol and they will quickly be history if you get carburetor cleaner on them.
Hope this helps. Have fun and stay safe out there.