Today, I returned with the paperwork, the old sticker, and the receipt. Today's clerk (as opposed to yesterday's clerk) did not want to look at the paperwork at all, although I had to fill out a new copy of the sticker application which she then taped the bits of the old sticker to before charging me the five dollars for the transfer.
Reading the application as I was filling it out, I saw that it called for the old sticker, but not any paperwork. Ah, well.
Meanwhile, while I had been standing in line waiting, she had informed a fellow trying to register his new car that he had to complete the registration within 30 days of buying the car to avoid being charged a penalty fee, for which he would need the paperwork showing when he had bought it. Interestingly, the sticker application said that the deadline for avoiding the penalty fee was 45 days after purchase.
There appears to be some lack of clarity on the rules here...
But the new sticker is now on the new car! :)