To make things a little easier let's use some more common names for the various gears of the diff. The driveshaft bevel gear is the pinion, the other bevel gear is the ring gear. The gears inside the diff cup are the spider gears.

It sounds like you're describing two separate problems. If you rotate one tire and the other tire rotates in the same direction that's an issue with the spider gears or possibly the diff outputs binding. If the diff outputs are bent they would bind on the ring gear or diff cup and cause the wheels to rotate in the same direction since the spider gears aren't rotating. Take the diff outputs out and roll them slowly across a flat surface to see if they're bent. Or simply try rotating each diff output in just the disassembled ring gear and just the dis assembled diff cup to see if there's binding.

The problem of the front wheels not turning when you apply throttle indicated a problem with the ring and pinion gears. The spider gears and diff outputs wouldn't prevent both wheels from turning, it would only prevent them from turning at different speeds. Shimming just the pinion isn't the most effective way to achieve optimal mesh but sometimes it's the only option. But when shimming the pinion it's important to make sure the pinion can move in and out of the diff case. Sometimes it's necessary to place shims on the outside of the diff case between the outer bearing and the drive cup to prevent the pinion from move in and out. After shimming did you check the mesh around the entire ring gear? Was it Notchy at all? Any play?