I agree with you about the crowds, but you'll find those are most obvious in Upper Antelope Canyon. At 100 yards in length, with a flat, easy trail going through it, it's the part of Antelope Canyon that pretty much anyone can manage. Lower Antelope Canyon is a different story. A longer canyon that requires more physical effort in the form of stairs to climb and boulders to navigate around, you won't see as many people there. Here's a video that depicts that ->: **SPOILER ALERT** This video features a full walk-through, so if you don't want to spoil the "surprise, forward to the 2:00 mark and the 18:35 mark to see the stairs http://antelopecanyon.az/lower-antelope-full-walkthrough-video/
Other ways to get away from the crowds:
Visit at "off-times," which are earlier in the morning and later in the afternoon. In Upper Antelope Canyon, you won't experience the "sunbeam" phenomenon, but the lighting will be softer and still beautiful.
Visit in the off-season, which is anytime between November and March. Because of the colder weather, there aren't as many people in either Upper or Lower Antelope Canyon. But, because of the tilt of the earth at that time of year, the sunbeams don't penetrate quite down to the floor of Upper Antelope Canyon.
Hope that helps. Have a good trip!