Thrift stores are great places to find quilted mattress covers for cage liners.
The cotton batting you purchased may be absorbent, but depending on how thick it is, it may take a long time to dry after washing.
I just made some liners in 3 sections that overlap and cover the entire floor of my cage. I used fleece on top fleece, the bottom is woven cotton fabric, and inside is a piece of beach towel. They work well and are easy to wash and dry,
My friend offered to sew tunnel pads for my guinea pigs and when I explained what it was for, she told me to buy cotton batting as the absorbent layer. I find the cotton batting thin and should have doubled up on it or buy polyster batting. As far as I know, there is no issue with shrinkage. I wash on warm or cold and line dry, which might explain why it did not shrink.