It does sound like contact dermatitis. I don't know if you know this or not but aspirin was first derived from the products of the Willow tree, and it is acidic which is probably why your fingers felt like they were burning. Since some time has passed you may now have atopic dermatitis or eczema which means your skin can become very dry and cracked like you describe.
You need to keep the area moisturized and moist. Sometimes low dose cortisone cream can make it worse. You might ask for a presciption cortisone mixed in eucerin cream, and apply 4 times a day, and wear white cotton gloves at nite (clean cotton socks will do) to help absorb the cream. Switch to Basis soap-a neutral soap-it is less harsh than other soaps. Moisture is key. I am giving a link to The mayo clinc info on this: http://www.mayoclinic.com/health/dermatitis-eczema/DS00339/DSECTION=1
I hope this helps you, and wear gloves when handling debris, you are more sensitized now to the sap. If you have not seen a dermatologist, but if you do not see relief from these measure, you might consider seeing one.
Take care, Paula
I am adding a new suggestion for udder cream-farmers wear by it. Southern State stores sell it. Paula