The only way to know what needle size will work is to check your gauge. You may get gauge with the suggested needle size or you may need to go up or down a size or more. How important gauge is depends on how important it is to have exact measurements. With a cowl you should have more "wiggle room" on the gauge than with, say, a sweater. A link to your pattern would be helpful.
When you change to a cotton yarn in a pattern written for wool, you may get a rather different result from what's pictured. That's not necessarily a bad thing, just something to be aware of.
Cheating is an option. . . . Cheaters never win and winners never cheat, but smart knitters who want to retain an iota of sanity do, cheerfully. ~~Kory Stamper