I have never ever fertilized lemon balm. It grows happily in my most terrible and spent garden beds. As a rule, most herbs don't require any fertilizer.

Bugs? Not all bugs are bad. If your plants aren't getting chewed to bits or damaged by the bugs, then consider the bugs benign and not something that requires killing.

Purple? Lemon balm turns purple when stressed by cold. Getting cold where you are? Got any snow yet? It's been snowing here in Western NY and even got down in the teens F. I didn't get much snow but 2 hours from here got 7 feet (no joke). Cold temperatures is a good probable cause for purple coloring and leaf die off.

FataMorgana

The only reason I've been fertilizing is because there are two plants in a rather small pot with potting soil that's going on its second year. I'll try cuting back on the fertilizing though, and be assured, if they were in the ground, I would let a mulch give them all the fertilization they needed.

It could be the cold. I've brought them inside so they aren't freezing, but my room seems unable to retain heat. The vents are open and blowing, but no warmth, and the plants are right next to the window, so even worse! Just one more thing to put on the "pro" list of getting a space heater, I guess.

Would the cold also make some of the leaves turn tissue-paper thin, slightly brown, and crunchy in spots? Either way, I'll try getting the temperature up a bit and see if that helps, thanks!

Lemon Balm is a tough hardy perennial. It is meant to die down at the end of the year and go dormant in winter. It is not suited to growing indoors. I don't think your plant has either pests or diseases. I just think it is physiological change due to the season and the inadequate conditions. It looks etiolated and I suspect that is due to insufficient light indoors. If you want to try to keep it alive inside it needs a huge amount of light. It will not need much heat. If it's comfortable for you it is probably too hot for Lemon Balm.