Update:

I took the above information to heart, and decided not to disassemble the things.

Checking out a few videos of people having a similar problem, I attacked the 7440 first. A little elbow grease on the glass parts with some alcohol seemed to help a little, but not all that much. Then I lifted the cover on the ADF and saw a few rollers and a couple of unpowered plastic rollers which , being spring mounted. apparently are used to make sure of good contact with the actual feed rollers. They were filthy . I'm guessing whoever owned it last used to feed a lot of material he had just printed through the thing, and by now a lot of toner had flaked off onto the rollers.

So, I removed them and with hot water, Dawn, and a toothbrush, I soon had them like new. So back into the machine they went, and time for a test.

Perfect! Absolutely pristine scans!

The moral of the story - printers and scanners are electromechanical devices. The electronics couldn't care less about dust, insect droppings, and the like, but the mechanical parts do. I've always taken the time to go over my printers to make sure they were clean, but I hadn't thought of the scanner.

Anyhow - a success story, and one happy camper here.