Hello,
disconnect the negative terminal from the battery. Using a test probe or test light, touch the cable with the test wire. Touch the negative terminal at the same time with the probe. The light should shine brightly and then fade. If the light remains on, you have a current leak somewhere. You can get a volt meter and test all the circuits one by one in the fuse panel. That should lead you to the source.
Also, the alternator should charge more than 13 volts. More like 14.0 -14.5 or so.
I had my battery go out (new battery) a few weeks ago. Had to have the car towed home. Next day while checking my connections I discovered that the ground wire to the engine block from the alternator was loose. If I had known at the time, I could have finger tightened it and it would have been sufficient to charge the battery. As it was, being dark, it didn't occur to me to check the connections when the battery went dead. So, there ya go... mechanics 101.
Art C.
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89-740 GLE;91-940 Sedan ;98-V90 Wagon ___ all running well