One thing most people don't consider is this- the overflow tube in the problem carb may have a crack in it. This would drain gas out constantly, as there would be nothing stopping it. Take the bowl off and fill it to just under the top of the overflow with gas, and see if it leaks out. If it does, either your drain screw needs replacing, or the overflow tube is cracked.
A guy with a steady hand and a good soldering gun (25W min) could solder a known leak shut, with care taken to not solder the WHOLE tube shut. Others have suggested pulling the tube out of the bowl and replacing it, but finding one with that funny crimped top and tiny diameter, much less getting it in, would be nigh unto impossible. Best bet, if the overflow tube is the culprit, is to cast about here or on the 'bay for a replacement.