If it is a new washing machine there are breaks screwed into the back of the machine.

They are put there so the drum does not move during shipping from the factory to the customer, maybe they were also put in when you moved. If they remain in the washing machine motor is fighting against the break pads and the machine will shake violently. Please look at the manual or search on the internet how to take them out.

Sounds to me like the drum might be floating really loosely inside the machine. It not spinning dead center is gonna cause the weight to be thrown around and move everything.

Are the feet set properly? They usually screw in and out. Try rocking it and see if you can adjust the feet lengths so it doesn't rock when shaking it.

Is it a side loader? The drum has springs that connect the drum to a weight in the back. Check if all of the springs are connected. If one is loose, the machine will vibrate like crazy.

I have a top loaded Samsung washer, and just replaced its four spring rods, which really helped with the shaking and bouncing issues.

What if you carefully distribute the weight of your laundry evenly? Mine doesn’t have an agitator but whenever I’m loading laundry, I make sure to do it in the shape of a donut, evenly distributed around the drum so that it doesn’t get thrown off

It's a new washing machine?

i had the same problem with a front-load machine. Feet seemed to be even, but the machine pivots clockwise about left front foot causing it to screw in (and shorten), which then worsens as it goes. My fix was to lengthen that foot more so that it seems to bear more load than the other three feet. I don’t consider it a permanent fix because that leg still manages to eventually get shorter anyway, but takes much longer.

I guess my next move is to drill a hole in a circular piece of wood giving a donut-like shape, and use that as a spacer to stop that foot from shortening under vibration.

You're filling it with too much...or not enough

-Chief Wiggum