I sweat a ton and use Tourna Tac even during the summer. I don't get the slimy comment because they don't feel that way to me at all but different strokes... Are you sure you're removing the outer plastic cover before applying them?
During the rest of the year I get a couple of weeks of play out of each grip but in the summer I have to change them out probably every other time I play. I previously used regular Tourna, Wilson Pro, and SuperGrab tut I feel I get the most mileage out of and have the best feel with Tourna Tac.