I'm surprised hobby quantity cutting oil is causing a problem, so maybe it's something else.
In the UK at the moment Hayfever, colds, and a new COVID variant are doing the rounds. (Don't panic, the COVID is a mild variant, only causing cold/flu-like symptoms and most of us are still vaccine protected.) And, rarely, some individuals are sensitive to the chemicals. Short or a rebreather, masks are unlikely to help because they catch particles, not fumes.
Unless a lot of hard cutting is done, cutting fluid may not be needed at all. Brass and cast-iron are cut dry, and I cut Aluminium with occasional brushed Paraffin, or sprayed WD40. Cutting fluid is reserved for threading, in tiny quantities.
The main purpose of cutting fluid is lubrication. It helps cool the tool too, but getting it too hot causes nasty fumes. In quantity, a flow of cutting fluid washes swarf away, so it isn't minced under the cutter.
For bulk cutting suds are often more useful than cutting fluid. Applied in quantity, suds cool and wash swarf away. Lubricates too, but that's secondary. Suds is an emulsion of a little soluble oil in water. Water is cheap and an extremely effective coolant. The soluble oil provides lubrication and prevents rust.
In the good old days suds was a serious health risk because nasty bacteria grow in soluble oil, and – before antibiotics – getting it in a cut could end in amputation or fatal sepsis. And it stinks. Modern suds contain disinfectants, but they don't last forever.
As flood-cooling is messy I avoid using it unless a lot of steel has to be machined. And carbide inserts reduce the need for cooling somewhat because carbide can be run hotter than HSS.
Industrially, mist systems are most likely to cause health problems, for which reason they are operated inside a sealed cabinet. Last time I read about it, industry (machine centres) seemed to be moving towards compressed gas systems. Something cold like liquid Nitrogen is sprayed accurately into the cut: the freezing gas blows swarf away and keeps the carbide cutter cool. There is no lubricant. A big advantage is it keeps swarf clean, greatly increasing the scrap value.
How much cutting fluid are you using. Try reducing it dramatically! As for advice if cutting back causes different problems.
Dave