hey dude I have expirence with this. I have solved this problem on the last 2Expresses that Ive owned. I have a 78 and a 81. Both have sat for a long time without gasoline in them and rust has formed in the gas tank. When I put gas into the tank that broke up the rust and moved that into the pepcock and into the float bown in the carbureator.
The resolve to this problem was fun for me and I wouldnt mind helping you with it. First you will have to take off the fuel tank and carbureator. dump the contents of the fueltank out and put some rocks gravel into the tank and shake the shit out of the tank and that will knock the rust loose from the inners of the tank. then wash the tank out with water. when you have the tank cleaned it is a good idea to spray it down well with WD-40 or something to keep any more rust from forming. Now it is time to check and clean the carbureator and the hoses.
I suggest replacing all the hoses with clear fuel line that you can see any sediment throught but remember to ask someone at the hardware store if the hose that you select is made for fuel line. some hose is not ment for that( I made that mistake once it sucks).. Anyways you will have to carefully remove the float bowl of the carbureator and extract the sediment from the bowl. this crap is allowing a slow trickle through youre fuel system and youre bike will start like a champ and run for 20 sec but then it will run out of gas. I like to use the tooth brush and carb cleaner method. when everything is clean then slap the bitch back togeather and youre scooter will probably run and idle the same only it probably wont die like it was before.
I bet that the former owner had set the idle up thinking he knew what he was doing, also thinking this would solve his problem so you will have to set the idle the way you like it.
I like to run Sea Foam through my engines when I first start working on them, but you first have to get them idiling..
Anyways I hope that I have helped you and if you need anything please email me with the address I supplied above.