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, Low Density, Low Elasticity
Suave, V05, ACV rinses, CO, and Gelatine PT.