Thank you for all your words. Have to admit I am not the most mechanically inclined and this is all new for me. Attached are a couple of photos so people can actually see what I am talking about as I am sure I am not wording things to well:
Note the small pipe on the bottom, back of the carb coming out. No plastic tubing on it (lower right 1/4 of pic)
If you look directly to the right of the screw attaching the air box to the carb, another small pipe coming from the carb, facing downward. This pipe as well has no plastic tube attached. This is the one that the gas leaks from after I shut the engine off. If I restart the engine and turn it off again, leaking stops.
Thank you again for any input, you guys rule beyond belief...
The pipe in the first pic is the carb drain tube. Gas only comes out of that one when the carb drain screw is unscrewed.
The pipe in the second pic is the carb overflow tube. Gas only comes out of that one if the carb bowl gets so full of gas that it reaches the top of the bowl. The tube is there so the excess gas has somewhere to go (out onto the ground). Under normal circumstances, the carb shouldn't overflow. Overflow problems are usually caused by either the float itself or float needle (sticky float, cracked float, crud stuck near the face of the needle, worn needle, dirty needle, etc.)
So can I still get a new float and needle from my local Honda dealer or is it difficult because of the scooter's age? If they are available, how difficult to install? I am pretty much a moron behind a wrench but willing to try if not insanely difficult...
Cold grinding grizzly bear jaws, hot on your heels...
I think you can still get them from honda.Just take the old one with you when you go. The float needle is under the float inside the carb. Its basically what the float bounces on or floats on when carb is filling up with gas.It allows the proper amont of gas in the carb. You have to remove a pin like device which goes through two tabs on the float. You slip that out and the float comes out. The float needle is right under the float about where it is attached by this pin. Just make sure you see which way the float needle is sitting in there when you take it out, so you do not put the new one in upside down.
Sorry the pics are so large... I am pretty much as good on the computer as I am working on my scooter. 
