Just wanted to post an update and once again thank all those that replied and offered their help and suggestions.
I followed Kerry's advice and ripped and burned a different DVD and got the same results - pixelation and freezes at the end of the disc. I then went back and played some DVDs that I'd backed up, but hadn't yet watched, and was disappointed to see that those, too, had the problems towards the end of the discs. Over time, I've noticed that almost all of my successful burns end at around 14 min. My burn logs show that the coasters took at least 15 minutes or a bit longer (burn speed was set to 8x for all). Interesting.
I then opened a new pack of TY's and reburned the first disc in which I'd noticed the problems and got a perfect backup. This led me to think I'd gotten a bad batch of TY's, but upon burning a second disc with the new batch, the problem showed up once again - always near the end of the disc.
My next step was to uninstall my programs (DVDFab, DVDShrink, and ImgBurn) and reinstall fresh copies. I then ran Defrag (took 6 hours) and when I got up this morning I ripped, compressed, and burned on a cold computer (I always do this since this laptop gets rather hot during burning). This time, I took Kipper's advice and set the output in Shrink to 4300MB, and used my cheapest disc in order to confirm it's not a media problem, but rather something going on with my burner or software. I got a perfect backup and didn't really notice too much decline in quality. (Thanks for that advice, Kipper!)
My routine for backing up includes always starting with a fresh 'restart' of the computer. I disable my AV and malware detectors and any other programs not necessary to be running in the background. I wait until the HD light stops blinking and I'm always careful to handle fresh discs in a manner that no part of the burn surface is ever touched. Everything I know so far about burning successful backups I learned here at CDFreaks and, as I've said, until now, I've had hundreds and hundreds of successes with very few coasters, so thank you to those that take the time to offer valuable advice and instructions.
So far everything points to a dying burner (at the least, an unstable one) or perhaps it could be a HD problem. I have a few other things to try (maybe cleaning the drive with one of those disc cleaners?), or perhaps picking up an external burner to test another theory.
BTW, I've found this is a very common problem. If you google 'backup pixelates and freezes at end of disc', you will find many posts on various forums (even here), and invariably, the advice is generally that the culprit was bad media. In my case, I don't think that's the problem. I'll be getting a new setup soon and may be back to see if anyone might be so kind as to walk me through installing a really good burner (I hear it's not that difficult), so I'll be doing some research into what would be a good choice.
Thanks again for everyone who replied:)