Welcome to the site, the best thing you can do to help yourself is to acquire a service manual. Either a Honda factory or even a Clymers (which are pretty easy to get on ebay). The reason I say this is there is a lot in the process to get to the carbs (taking off the tupperware can be a pain). As far as the fuel in the airbox & hesitation goes. It could be a number of things. You did not tell us what year your 1500 was, some of the early years had hestitation issues.
Can't really address the fuel in the airbox issue but it isn't uncommon for the 1500's carb to have the jets on either one side or both plugged. The 1500 has what resembles a 2bbl carb with the right side serving the right cylinnder bank and the left serving the left bank. We've seen that here in our shop several times where one side was plugged off. I don't wrench but it's what my techs have told me