I make many scrap quilts for a home where homeless people sleep. To save money I have used mixtures of poly, new cotton fabric, sheets and even some jersey all in one quilt. I do use a lot of cotton sheets, and I have mixed Batik's with cotton too. I use Dream Green batting.
Here is what I want to say. I wash everything for shrinkage first. This way the cotton piece won't shrink along side of a poly piece. My quilts all come out great, and are sturdy and washable. The poly batting doesn't shrink either, so my quilts are stable. (I even use new quilting cotton along with the poly etc. but wash it for shrinkage first.)
About shrinkage. I learned from an expert. The shrinking takes place in a hot dryer mostly from the POUNDING. A hot steam iron won't shrink the cotton. So, I put new cotton fabric through gentle-hand wash with no detergent (to save money.) I just use the rinse cycle. Then the most important is the hot dryer for as long as it takes.
Of course sheets are laundered with detergent -- the whole nine yards.