I heart these new bibs! They are chenille on one side and just plain cotton on the other. I debated putting anything between the layers, but they are pretty thick and absorbant. And they are a good size for a toddler.
I've sewed a lot with fusible fleece and I have never had to do anything different. Maybe just try fusing a piece of scrap fleece to your scrap fabric and then play around with the tension to see if that helps.