re: "...Shutting it down, the voltage starts to fall measurably. I believe that this indicates a failing battery...."

No. It's normal for a battery's voltage to fall immediately and abruptly upon shutting off either the engine (viz., alternator) or an ordinary charger* on it. While the battery being charged should be at ~14 volts (higher in cold weather, lower in hot weather -- yes, it's very temperature dependent), the charge should drop quickly to about 12.6 v and then stay there, in healthy battery.
If it drops and stays at a lower (than 12.6) voltage, that could indeed indicate a problem -- but it's this steady voltage AFTER it dropped that's important, not the observation that it didn't stay up at 14 volts after charging stopped.

[ * If you have a multiphase battery charger, then it should initially charge at ~14 volts but then eventually change the charge mode and drop to and stay at about 13.3 volts indefinitely. But as above, once the charger is shut off, the voltage should drop immediately as I described above.]

Other than load testing with a load tester, the best way to check your battery is to check the battery voltage while cranking the engine. If the battery voltage drops below about 10 volts during cranking, you have a bad battery.

Also, age is the poorest criteria to judge a battery. I have seven year old batteries that are in good condition and start my cars in cold weather (e.g., it was -5 F this morning in NJ), and yet batteries just a few years old can fail. It's mostly all about how they're cared for -- e.g., making sure they're charged up if you leave the car for a while (weeks); never leaving them to drain the battery with the lights on; etc.

Good luck.