I have a 2002 650 Savage and it's a interesting bike. I had to put a clutch in it shortly after buying it, two years ago. The bike has 28,000 miles on it. The only real problem I found is you can't shut off the fuel to the carb. I had the float stick and fuel came out the over flow tub onto the exhaust and catch fire. It melted the brake light switch and the wires to the natural safety switch. I was lucky I had a water bottle handy. From reading I found that sticking float isn't uncommon and I read a bunch of horror stories about rebuilding the carb. I use a super carb cleaner and then tapped the carb with a hammer and freed the float. It hasn't given me a problem since. I have since installed a real fuel shutoff valve inline. My only other complain is the seat is not good for long distances, I can only take it for about two hours. There are after market seats out there that are said to be better but instead of spending the money I just bought a 1986 Goldwing as a touring bike. My advice is run carb cleaner though it from time to time and install a shutoff valve in the fuel line. Have fun.
Sounds like your carb bowl drain line is too short and mis-routed. It should not be dripping onto the pipes, is routed down and stubs beyond the frame by the fender.
Also, did you have the petcock in the "PRI" (Prime) position? That bypasses the vacuum pull off valve opening mechanism, helps to fill carb bowl when not run in a while and all the fuel evaporated from the carb.
The bike has a vacuum line to the tank stopcock. When engine is running, a vacuum pulls an internal diaphragm in the petcock and opens the valve. Inasmuch as I don't like this set up, prefer the old mechanical Main-Reserve-Off type petcock. While I owned the bike, I installed 2 new factory petcocks. Seems they'd only last a couple years, then the diaphragm would fail and I'd have to run the bike on "Prime" until replaced.
Before I passed on the bike to a friend, I bought a non-vacuum operated manual replacement for an older Suzuki or Yamaha, can't remember which model. Gave it to the new owner. Only thing is the Reserve pick up tube is slightly shorter than the LS650's, so you had like a half gallon instead of 0.8 gallon as reserve. Helps to carry one of those gallon saddle bag fuel jugs, so you're never short. Out here in rural NM and Panhandle TX, the village you're passing through may have only 1 gas stop and it may have limited hours. Also, I've seen where the fuel truck delivery was not timely so they were out of fuel. Another, I've seen where they had a pump motor failure or power outage. Gas sales were closed.
Wouldn't hurt to add a little Seafoam carb cleaner to the tank every couple months. I learned this with my 4 carb 2001 Kawasaki ZG1200 Voyager XII tourer several years ago. Fuel these days has Ethanol, tends to gum up carbs. Bike was idling a little rough in Oklahoma City. I put the recommended shot of Seafoam in the tank at next gas up. 200 miles later by the time I hit Oklahoma's southern boarder, the engine was running smoothly.
Regarding float problems, I replaced the float valve on the Savage, which cured the problem. Wouldn't hurt to add an in-line fuel filter too. Also check the factory fuel screen in the petcock. If tank is a little rusty, it may have some crud restricting fuel flow or a small piece passed through screen and lodged in the bowl needle valve area allowing fuel to continue to flow when it should have not.
Anyway, my 2 cents on what I've learned.