I had exactly the same problem when I first bought my Trek 1500, and I weighed about 230 lbs, with both Bontragers and Contis. Flats were never consistent, and happened almost every ride. I was beside myself with frustration.
Finally landed on Michelin Carbons with Michelin Air Stop tubes, and now ride 1000s of miles without flats.
Still don't know if it was the tires