Since it’s oil based, just wash any areas with DAWN dishwashing liquid, using a terry cloth hand towel (it needs to have a texture to work). Repeat this 3x.
No more reaction.
As someone who has had a severe reaction my whole life, finding this out was a God Send!
Another way to handle it is to wash with mechanic's hand cleaner, the semi-liquid stuff that usually comes in a pump bottle. Again intended to remove oil, and somewhat better than DAWN for this.
There are two kinds, the slimey kind and the gritty kind. The gritty one works better for me.