Analysis about H bridge to drive DC Motor
Here's a reason to get rid of D1 and D2

With D1 and D2 in place you are going to eventually destroy the lower MOSFETs because the flyback diodes in the upper mosfets have nowhere to discharge the flyback currents - Normally flyback current from the motor will find a path to the positive rail and slightly charge the capacitor you have on that rail. D1 and D2 are not going to allow this without something going wrong.

What might happen when one of the lower transistors turn off (say U2A) is that you get the flyback voltage as mentioned above and this might cause Q3 to turn on and this will turn U2A back on again. This could be your problem.