I am pretty sure that coco and cocamidopropyl betaine can get rid of cones. So you should be able to still use cones and a low poo instead of a sulfate.

However, it could be the cones initially that are adding to your irritation, like the others have suggested.
High Porosity, Fine, Thin Density, Low Elasticity
Natures Gate hemp for co-washing, detangling, and leave in + CO.