Sewing multiple layers of fleece - stretch / fabric drifting / bunching HELP

[ItWasGoodWhileItLastedRVN123]

I'm currently sewing something that involves a couple of layers of polar type fleece with a waterproof and also an absorbent layer in between (guinea pig cage liners!).

My problem is that despite cutting out the shapes (mainly rectangles and squares) perfectly and pinning together, by the time I sew an edge, the top (or bottom fabric) has pulled and drifted (or maybe stretched) so much that by the time I get to the bottom there is a good couple of inches of overhang of one of the layers of fabric!
After research, I bought a cheap "walking foot" from Amazon (the branded Brother one for my machine was £38) but it didn't seem to do anything special, the problem still happens (not to mention it broke today after only one day of using it!).
I've now bought the real Brother walking foot and am waiting on it arriving, but what else can I do?
I'm pinning everything within an inch of its life in two rows of pins, but the fabric drift is still happening.
I'm switching to a ballpoint needle tomorrow when the arrive, will this help?
I also read that the presser foot pressure should be lighter when sewing with fleece but there is no way to adjust this on my pretty basic machine.

What else can I try? It's driving me mad!
Any help would be greatly accepted. Thank you.

I find this a real problem too.
I do a bit of patchwork, and find that quilting the layers is so tricky.
I had some success with the walking foot- but not a massive difference.
I try to make sure that I have as much space as possible, I never try to attempt any straight/ measured lines, and tack layers.
Still all wonky though.
How do people get such neat layers?

rack with small stitches. It's time consuming but so worth it

*should be tack

Are you pinning in line with your seam or at 90degrees? I find things shift less if I pin at 90degrees.
You could also try hand basting, or use an adhesive spray to stick the layers together before sewing.

A lot of time basting.

Or use a craft glue stick - water soluble - and throw away the needle afterwards.

Or the proper 505 spray

Or safety pins.

But normal pins will shift too much. And even with a decent walking foot, those layers will shift.

Try sticking the fabrics together (by glue or double sided tape), then hold all 3 together, by the edge with one hand when you feed it through the machine. With your other hand hold the end of what it is you are stitching and constantly make sure it is still level. Stitch at half the speed you normally would. Don't feed it through the machine, let the machine do the work.

Sorry contradicted myself a bit there. When I mentioned holding the fabric and feeding it through the machine I meant to say guide it through as in, making sure the foot moves it along, not pushing the fabric through.

Bulldog clips.

The heavy duty ones.

I'd overlap the waterproof fabric around the fleece and liner and then sew. It will keep the fleece stable.