But I forgot to cover how I fixed the stability issues.
Testing the corroded traces around the battery, I found that DRD9 was open between pin 46 of Denise and pin 5 of Agnus.
That's one of the data bus lines; it's surprising anything worked at all! Restoring DRD9 (white alligators) fixed the vertical lines as well as the frequent crashes. Woot! 🎉