A special thank-you to everyone who kindly responded. This problem has been driving me nuts for the last month or so. Following several people's advice, I put the imported hide in a heavy plastic bag and set it aside. I also threw away any remaining scraps. (All this with gloves on, of course.) Then, I did a little work with Live Oak (American origin) veg-tan. We'll see if the blisters, as I suspect, completely clear up. I also took Johanna's advice and washed my hands more, not less.
Also, as 12stones mentioned, I've fallen out of the use of gloves with leather dye since I'm using the Eco-Flo water-based stuff and I try to be careful. Nonetheless, I'm going to start dyeing with gloves on again, too. Might help.
Thanks again, everyone! Regards, -Alex