Wanted to thank everyone who chimed in with advice. I thought you all deserved an update: I ended up taking it to a professional who – I now know correctly – surmised the engine is, after 110,000 miles, essentially just very tired, specifically tired of eating unleaded fuel and low-zinc oil.
I also thought some of you might be happy to hear I am, as a result, putting the Six through a complete top-flight engine rebuild!
Met with the blueprinter yesterday to do some disassembly; we came across issues like (and forgive me if I bungle some terminology, here, my engine expertise is a little rudimentary) worn valves/valve stems shimmying in their guides; oil pooling on the back side of the valves through the guides; eccentric, excessive wear on the fuel pump lobe on the camshaft; concave surfaces on the valve lifters; a very slack timing chain; hard ridges in the top of the cylinder; completely glazed cylinder walls; and a cracked top piston ring with a chunk missing (on #2 cylinder; we only did the one while I was there).
The engine will be getting a no-expense-barred as-close-to-factory-spec-as-possible rebuild with the goal of keeping it original-appearing. To make it extra-difficult, we've even decided to keep the engine patina. The crew I've hired is accustomed to refreshing survivor (mostly Hemi and AAR or T/A) cars, so enjoys a challenge like this. Suffice it to say, at the end of the day, the cost of the rebuild will be roughly that of some complete Slant Six cars

But I am a loon, love my car and plan to drive it 50 years on top of the 50 it's already been through, so I figure it's earned it.
I plan to put together an article chronicling the rebuild when it's all done, and will be sure to share it here once it's written.
Thanks again, all, for your help and support! It's much appreciated.