I tried it with the airbox off an same results....so not sure that matters much at all.

The hesitation in off-idle throttle response is due to being too lean (when you whack it open from idle, it sits there).  The longer it waits, the leaner it is.  First thing to do is remove those brass caps over the mixture screws.  There are write-ups on this proceedure.  I'll go search for 'brass caps' and post back what I find.  If you don't want to wait, or want to search other words, or the FAQ, go for it.

http://gstwins.com/gsboard/index.php?topic=5469.msg44610#msg44610
http://gstwins.com/gsboard/index.php?topic=13148.msg110842#msg110842

not to speak for Praefex, but my bike only hesitates (boggy in low rpm) when it's cold.
After a min or two both cylinders fire fine and she runs like a champ.

how can the fuel mixture only affect the  bike when cold?

Without actually getting my hands on your bike, I'm conjecturing that the temperature constricts the idle circuit holes enough to restrict flow.  Did you already do your coil test?

And, is it really a hesitation (too lean, bucks back and forth), or bogging (too rich, engine looses power/just slows down/engine burbles/makes burbling sound)?

At this point it sounds like it's just normal, cold-blooded action of a cold, air-cooled engine that may be still too lean down low.

I don't know about the coils just yet.  I will test them when my new parts come in.

speaking of which I bought new coils for the bike from ebay.  When I put in the new clutch cable I will swap the coils out.

If the problem disapears then it was electric if not I am just going to tear into the carbs again.

------------
The engines definitely bogs down as you suggest.  I am positive one cylinder is not firing
(whether carb or electric is beyond me) if it is too rich it is only one carb. I am very hesitant to adjust the mixture screw, seeing as how I have not "modded" or "rejet" the carbs, I simply tore them apart and cleaned them.
I under stand how it works but the bike runs like a champ after a minute or two of boggy warmup.

The mixscrew will not "only" adjust the warm up mixture.

I am placing my bets on a bit of rust or something clogging something that only affects it at warmup.

I hope you are right and its just normal. But she didn't do it for the first two years i've had her.

Thanks for the help. 

Hey PRAEFEX  any luck?

GS is WORKING!!!

Did one more super thorough carb clean, drilled out that brass air/fuel caps adjusted it all down and the bike works like a charm now. I reaplaced a few o rings and such. I think my next project is to tackle the fuel host routing and clean up that abortion with an after market solution.

Thanks for the suggestions everyone. I still have some coils on order too so I will install those as well, I'm sure they can only help at this point.

Bought a '91 that had been sitting for a year or maybe a little less.  It bogged down like yours.  I did pull the carbs and cleaned them with throttle body cleaner.  I also poured 5 ozs. of a product called Seafoam in the gas tank when I filled it up(1ounce per gallon).  The Seafoam stuff is pretty good stuff and is used alot on the Toyota Celica Forum for cleaning out the system.  Also drained the old gas and put in fresh.  New plugs and air filter.  With those 5 things done it runs like a champ!
I bought the Seaform at Autozone for around 6 bucks a can.  I'm a believer.