Oh, very sorry about the wool allergy! Maybe this idea will help...
A couple of months ago, I purchased one skein each of the Lion Brand organic/recycled cotton yarns not available at Joann's. I also lifted up and "checked" (tee hee) the Lion Brand cottons that were available at Joann's.
One of the yarns that I ordered, "Recycled Cotton," has some "grab" to it. 74% recycled cotton, 24% acrylic, 2% other fiber (not wool!). It's a "medium" (Class 4 weight), recommended for size 8 (5 mm) needles (dbl strand: prob. #11).
They have three or four others, but of the ones I've held in my own hands, this one might be the most helpful, if you can find a color that will work for you.