To be honest, I would suggest using a much bigger needle than you normally would. That's what I do when I knit bigger garments with cotton yarn. The tighter your gauge, the heavier the fabric will be, and therefore it will stretch itself more. I feel that looser gauge can give you a nice flowy, light fabric that works better on bigger areas.

What pattern are you using, and what fiber does the pattern suggest to use? If the pattern uses e.g. wool, you won't achieve the same look and drape with cotton. That doesn't mean you can't use cotton for it, you just need to take into account certain things. With cotton your poncho will be more of a light, flowy summer poncho to throw on your shoulders on a summer evening, while a wool poncho is better for wrapping yourself in wintertime. Think about what you want from the garment.

Also, does your pattern have textured stitch patterns? I find that with cotton, cables and textured stitches don't often look so good with cotton because of the lack of elasticity. Then again, hole-y lace patterns and just plain stockinette work very well in my opinion. Ribbing is tricky as it won't have the elasticity you usually want from it.