Nikkilu - have you thought about washing your red flannel sheets in "Retyne" (Retain), that will probably stop the bleeding, you can purchase from your local quilt shop.
I have used fleece sheets on the back of my quilts - then the common comment that I receive is "that is the best blanket ever", I use fleece sheets on all the beds in our home, and also in the RV - year round, it is simply amazing how well we sleep now, since changing to fleece - even in the hot of the summer - they breath!!!
I have also used Egyptian cotton (better quality muslin), very close weave and very soft, you might want to look into that for your quilts.
I purchased it at our LQS.