unless you are cutting on the bias you should not be getting alot of stretch from even-weave cotton fabrics...
if you are not cutting on grain there is always alot of stretch regardless of fiber content.
what or when is the fraying happening? on the cut edge before sewing? your seams? pre=washing?
you always get some fraying when you pre=wash but once you trim up the edges you should not be getting alot of fraying..
except home-spuns and flannels which have a looser weave and fray easier.
the tighter the weave the less fraying of cut edges.