2006 Ford Explorer Uneven road driving problem

Hi, I have ford explorer 2006 Eddie Bauer 4.0 ltr car. When driving over bumps (or uneven road) with speed over 40km/hr or accelerating over bumps, the car makes sounds like a heavy truck with a resistance to wheels while accelerating. Also, the car is trying to deviate to right rather than going in straight. The trasmission hardly shifts especially to first and second gears while the rpm increases higher than usual driving. The car goes back to normal if I stop then restart it again. I went to different shops and mechanics but they could not fix it. What should I do? Please advice.