Thanks to everyone who took time to offer advice. I have resolved this problem, so I'd like to circle back in case anyone out there comes across these posts while researching the same problem I had.
It turns out that, for just playing the DVDs, they were not as damaged as I had assumed. By merely ejecting the disks, and trying to play them a second or third time, I got different results. My initial impression that, once a disk messed up at 17:45 it would always do the same thing at the same place. This was incorrect.
Before I discovered this, I had a very unsatisfactory exchange with Lite-On customer service. They e-mailed me a list of different types of discs and suggested recording speeds. The subtext was, "In the future, do this and you will be OK." Really? Thanks! They offered no guidance on how to copy DVDs which are going downhill.
I looked into buying another DVD burner and went to a good store near me named Micro Center. While I was there, after talking with a cs rep, I realized that I should try screen capturing, and then store the .mpgs on different hard drives.
They had five different brands of capture adapters. I looked at the Dazzle HD, regular Dazzle, and three cheaper models. They all only had the yellow RCA video plug, and I asked the rep why they did not have the green-blue-red "component video out." I did not follow the answer except that "they don't make them that way because it wouldn't help."
After mulling things over, I decided to roll the dice on the cheapest adapter of the bunch. I paid about $25.00 for a Sabrent adapter which came with Ulead VideoStudio SE software. Getting the audio set up correctly was a real pain (that is one of the things I have never gotten down pat in XP.)
The Ulead software baffles me, but I don't need to do much editing. I just played the DVDs, set a timer, returned to my computer and hit "stop capture" at each movie's end.
The results of all this have far exceeded my expectations. I took one of the .mpgs which I captured, converted it back to DVD and burned it. I then put it into a DVD player and compared it to the original DVD. I even hooked up two DVD players so I could flip between them to compare. They were indistinguishable!
Considering how much those old films meant to me, I am much relieved, and I have learned my lesson in regard to storage media. Bkup, Bkup, Bkup!
In the next few months, I will look into burning all of the .mpgs to DVD, but for now I'm done with this mess. It's been a frustrating roller coaster ride, and I'm glad this is over.