Q: SUV tries to accelerate and stop at same time

My SUV acts like it wants to keep accelerating when I press on the brake pedal to slow down. It has a jerking back and forward which causes a hard stop. It seems to want to accelerate and brake at the same time because the RPMs continually and constantly move up and down switching from V4 to V8 mode. The same jerking occurs when going up short inclines and also when driving downhill, forward and back until I completely stop. When starting and staying at a low speed of under 35 mph, my SUV can not maintain the speed; it wants to stop. It never has died fortunately, but I have to punch the accelerator for it to pick up speed. Recently I was in a slow-moving line, coasting about 10 mph and my foot wasn’t on the brake or the gas pedal. The vehicle kept moving forward on its own. The problem seems to always happen at low speeds from 0-40 mph. Every time I go to the dealership, they can not duplicate the problem because this problem is intermittent. This could be dangerous if my car won’t stop and causes an accident. Can you please give me your expert advice of how this can be solved?