My alternator behaved a bit like described. It was a bad alternator - I think the voltage regulator was blown. Replaced it and my voltage sits at 13.7v, 13.3 with the lights on.

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1983 240D Silver/Blue "Sadie", unknown miles. 100k miles on WVO single tank, converted to 2 tank about 10k miles ago, FPHE, Injector Heaters, Aux Fuel pump. Alcohol/Water injection. Frantz oil bypass filter
1992 300D 2.5 Turbo Green/Tan "Jade" 267K miles Stock.
1992 300D 2.5 tan/tan "Rocky" on 2 tank WVO. Pressure actuated Turbo "rat's nest" surgery completed. 197k miles