Under the float covers is likely where the problem is (in the bowles and around the floats and jets). What do you have to lose by trying a little harder on those screws? You can get used carbs on ebay for $75. A mechanic is going to charge you that just to pull the bike into his garage.
eBay Motors: 2003 Kawasaki Ninja ZX6 E ZZR ZZ-R600 Carburators Carbs (item 270235047554 end time May-12-08 18:29:19 PDT)