So glad that you have joined us. I have asthma also and that is one of the reasons I switched to WS Oils. I also have wished that they would dry a little faster, so I have experimented with some of the additives that do speed up drying.
I like using walnut oil, instead of linseed oil which seems to both my breathing. I recently started using walnut alkyd which speeds up drying, but it definitely bothered my breathing. It seems to be alkyd and linquin(??) bother me.
When I want things to dry faster or use the alkyd for glazes, I try to paint outside if at all possible, and be sure to let it dry in a room which I don't go into too often.
Hope that helps.