Thank you both for your help. I appreciate your suggestions and advice. The project is finally, finally done!
This has been hands-down the worst DIY project I've ever attempted. The PB Blaster did a great job of loosening the nut from the underside of the sink, but did nothing to help me remove it from the bolt. I tried drilling the plastic and drilled several lovely holes but it never split. The plastic was oddly soft and would just mush away from the drill. I ended up buying a Dremel and cutting them off, but even that took several hours. It was terrible. The pieces I cut off are terribly corroded (I didn't plastic could look like that!). The old faucet had no plumber's putty, thread tape, etc. at all.
The new faucet went in reasonably well but the nut that holds the new faucet in place is so large I don't have a wrench that can fit it. Now I have to go buy a new wrench.
This project has cost me not just the cost of the new faucet, but also a new Dremel, a new wrench, my favorite pliers, two weeks of my time, and several trips to the chiropractor. It's just not worth it. I will pay a plumber anything he wants to do this, I am NEVER doing this again!!!