Transmission Not Engage problems of the 2006 Toyota RAV4

I have the 2006 Toyota Rav4 base model with an automatic transmission. I have noticed that the transmission is very sluggish to respond to braking, such that the engine sometimes continues to rev forward for a second or two even after the brake pedal is depressed. This leads to excessive braking distances. I believe this problem is the cause of an automobile accident that we suffered on 8/5/06. In dense highway traffic, I was about 20 feet behind an older Toyota pathfinder and had just accelerated from about 15 mph to about 25 mph, as the distance between the vehicles was growing. Then the pathfinder in front of me suddenly slammed on his/her brakes because another vehicle swerved directly in front of it. I slammed on my brakes too, but the Rav4 did not immediately begin to slow. Instead, I could feel the engine still pushing the car forward. The Rav4 finally did begin to slow down right about the same time the pathfinder came to a complete stop. As a result, we hit the now-stationary pathfinder in front of us at a speed of about 5 mph. Fortunately, no one was hurt, but the front end of the Rav4 was mashed in. I have driven dozens of different vehicles over nearly 20 years of driving experience, and I think every other one would have come to a complete stop well before the point of impact (as did the pathfinder in front of me, despite a vehicle cutting directly in front of it). This leads me to suspect there must be something wrong with the Rav4 transmission. The transmission should immediately disengage from drive whenever the brake is depressed, but there seems to be a dangerous time lag. When I took the vehicle to the dealership and explained what happened, they road-tested and examined the vehicle and said nothing was wrong. They claimed what I experienced was the vehicle down-shifting, which results in higher rpms but less forward motion. I don't buy it. I have tried jamming on the brakes to cause such downshifting and it doesn't happen.