On the very end of your shift linkage where it bolts to the bottom of the car there is a rubber bushing with a hole through the middle where the shift linkage goes through and flares out, it's the bracket that holds the shift linkage up.

Your rear motor mount is probably broken and you got on it causing the motor to rock forward yanking the shift linkage through the bushing, when your stopped the motor is resting where it's supposed to be, letting you change gears like normal. When your moving/going forward the motor rocks forward pulling on the shift linkage, not letting shift normally.

Your going to have to go under the car and remove the bracket 2 12mm bolts (I think) and try to put the bushing back on the shift linkage, then fix your motor mounts so it doesn't happen again.