The Oilers, Falcons, and the Lions all ran the run and shoot for years and ran it pretty effectively. All that yardage Warren Moon put up in Houston? A result of the run and shoot. Same thing when Bobby Hebert and then Jeff George were in Atlanta(they ran it while June Jones was OC, then HC). Scott Mitchell put up some big numbers and IIRC even went to Pro Bowl because of the run and shoot. It was a very trendy offense and lasted a lot longer than three years, as stated by Professor. Eventually, defenses just got faster and teams couldn't protect their QB or run the ball effectively in that offense.
I always joked with a friend of mine that the Saints ran the shoot and shoot offense last year because of how much they passed. I don't think they really used any real run and shoot offense concepts, though. We just passed a lot. We used a lot of receivers, but rather than just using 4 WRs, we used backs and TEs in tandem with our WRs. If anything, our offense was probably most similar to the Bill Walsh offense used with the 49ers when they went to their first Superbowl in the 80s. In that offense, a short pass to a RB was basically considered a long hand off.