I've learned a lot from these posts. Thanks, everybody. I've never thought about my TV. I'm a hi-fi guy, and I've thought a lot about signal fidelity in my stereo system...but never a thought to how the TV system works. I'm a lot more aware now.
I liked the suggestion about the splitter. Easy and inexpensive. And it brought me to the realization that these days I'm only using one TV, yet I have an old, el cheapo, three-way splitter in the signal path.
So, out the splitter came, replaced by a simple coupler, and, voila, instant improvement. Channels 3 and 47 are looking solid. So, a good solution for my situation was simple. It could have been obvious, but, like I said, I'm not used to thinking about my TV system.
Thank you for all the help.