Thick, greasy stuff is most effective. I have a prescription for something called Absorbase which is pretty much the same thing as Eucarin. I have no idea if these things have similar names where you are. Latex gloves - POWDERED ones are worst and there's some who feel it's the powder that causes the problems since it absorbs and holds stuff the latex off-gasses - can cause allergies, but that doesn't sound like what you have.

The number of times you wash your hands and the oil-dissolving capability of your soap may be causing problems. Also 'suddenly' indicates that something changed which may have led to the problems. Different soap, different gloves, different chemicals you work with which might permeate the gloves somewhat, or some procedure which allows the tips of your fingers, but not the rest of your hand, to come into contact with a solvent.

Anyway, go for the thick stuff. It will gunk up your strings, but I don't know what you can do about that.