True lanolin allergy is supposed to be fairly rare, but it does happen. The test is simple: try lathering up some MWF on your bare hands, with no brush. If there is no skin reaction, then it was almost certainly brush-burn.
Assuming it isn't an allergy, try some bowl-lathers or palm-lathers to figure out the right technique for MWF. I agree with the earlier advice to load up the brush.
I would try that on the forearm instead. There are things (like hydraulic fluid) that will not even faze my hands, but I will get splotches and irritation on my inner forearm if any gets splashed there.