The gauze should start to slide a bit easier as it becomes more broken in (with washing/use), but crinkle cotton is more textured that jersey, so it likely just won't slide as easily.
For hot weather carries in a long wrap I like a front wrap cross carry without the crosses spead. Just leave them bunched and cross under the baby's bum and tie in back. Don't spread the crosses over the baby's body, and you'll only have one layer over baby. Ruck carries are a good single layer carry if you want to carry baby on your back.
Gauze/crinkle cotton is definitely cooler for summer, but as your baby get older and heavier you may find you want the extra width/support of a heavier woven wrap.