It sounds like your brakes are not working correctly. They are supposed to hold the drum during agitation. It is unusual that they would not do so, but that seems to be the issue here.

You need to go in there and change out the brake package, but it will require a special brake tool (the brake spring is under high pressure and can injure you if you do not do this correctly). Maytag will no longer allow access to their tech site, so you need to contact them and see if they can get you instructions or a service manual. You can also purchase a manual on several web sites or auction sites.

If you click on your model number you will see a set of diagrams with exploded views of your machine.

I cannot tell you what parts you need, you just have to see when you get in there. If you order the parts from this site and you don't need some of them you can return them.