2006 Tacoma 6-cyl 4x4, and the battery is 3 years old. Truck would not start this morning, so I checked the voltage - 11.8V. When I cracked the ignition, it just clicked. To me that usually means a dead battery. I was able to start the truck with jumper cables. When I disconnected the cables and checked the voltage of the battery, it showed it slowly draining until it was below 10V and dropping...slowly, but dropping still. I turned the truck off, checked the battery voltage and it returned to 11.8ish range again. Dead battery or alternator?
What you are describing is your alternator being bad. You can try charging the battery separate from the vehicle until full (over 12.5-13 vdc), then try what you're doing over again. If you're seeing the battery continually lose voltage as the engine is running, this is indicative of a bad alternator.
Also, you talk about the battery returned to the 11.8 vdc range after stopping the engine. I don't know what the scientific name for this is, but I call it "rebound". You see it quite often in batteries of any kind which are under load, then once the load is relieved, the voltage goes up some.