When our washing machine is filling with water, the shower runs like someone turned on the water. Once the washing machine has stopped the shower head no longer leaks water.

Now that's new one. What ever the problem is it has to be in the shower valve.
The shower valve must be a single lever. What brand? The only logical explanation that I can come up with is that the washer must be taking all the water pressure off one line (hot or cold) and the pressure on the other line is over riding the seals in the shower valve because of the pressure imbalance. Now that should happen. What is your water pressure. It shouldn't happen no matter what the pressure but that is a place to start.
If the washer is using cold water, is the water coming out the shower hot or cold. Does it happen if the washer uses warm water. Does the volume or the temperature of any other faucet change when the washer starts? Is this something new? Has any plumbing work been done.