Petcocks and float needles are just part of the carburetor game. The good news is parts are typically cheap. This bike isn't any different than any other. They wear like any other, but they also can be eaten or gummed up faster if improperly stored. If you're comfortable working with carbs I would suggest acquiring (downloading perhaps) the shop manual and going through it regardless. As you said its the first of its kind that you own, you may not know if its running healthy or not and you may not know its history either. Not that I am trying to instill any fear, but if its running too lean/rich/not vaporizing correctly you could end up with engine damage in the long run. With your current situation if the float isn't sealing the gas overflowing would be washing down the cylinder walls causing excessive wear. Its no different for any bike, once again.
The drain screw shouldn't leak regardless. If you have gas leaking out of the middle hose at the bottom of the float bowl, well its leaking and pretty easy/cheap to fix. It may be a seal or crud or just loose. I usually place some paper underneath the carb after doing any carb work so it shows the "splat" where it dripped so I know if and where to look. Some paper works better than others, especially those that lose their colour when gas hits it. The two overflow lines towards the back of the carb suggest the float is leaking. That is more commonly the case. If that is, I would suggest doing a careful inspection first, and if in doubt replace it as a kit. I think I got both the petcock (with pickup screens, etc) and float needle set (housing, o-ring, needle, filter) for $15ea from boats.net. I posted somewhere in here that their sister site has a 15% off deal that may come in handy.
As for the petcock, if you carefully unscrew the faceplate on the side (I think its spring loaded, might be best to do when off the bike in a clean room) the diaphragm usually shows a line leading from one of the inlets to the outlet when it gets worn and/or eaten. Some gas is harder on it than others.
If you're pulling it apart, I would highly suggest taking a good look at the jets, emulsifier (can't recall its friendlier name) tube, and needle. Please read through the two carb stickies (cleaning and tuning). The cleaning one isn't directly applicable as a few things are different but the pictures help out a lot. The shop manual's diagrams will be more applicable.