Thanks everyone for your replies. I'll try riding a bit, and see what happens; at least until I get my new tires. It was just so damn nasty!
I very often carry my bike in the back of my van. If my tires are really dirty, I like to at least rinse them off before it goes in the van. This stuff is beyond nasty. It's thick, black, oily, sticky, goop. And, it stains everything it touches.
Anyway, I've got a new pair of tires on the way. I've got a couple of friends who would probably be happy to take my sticky ones. They're probably not as bad as I'm thinking they are. But, when it comes to my bike, I don't screw around. My bike is pretty damn important to me. I'll usually do what it takes to keep it nice. And, this vile crap on my tires is making me crazy!
RSNG