I would replace the battery, the lucky batteries make it 8 years but that's not the majority. Most are at 6 if they are a quality brand. I don't think that'll necessarily solve your problem either, but it's probably needed.
Did cars with alternators have Generator lights? Seems backwards to me but also seems entirely possible.
To find a draw, I pull the negative battery cable and put a test light between it and the negative post on the battery. The test light will light up if you have a draw. Next, start pulling fuses until the light goes out. Once it does, then you know what circuit is drawing (whatever the fuse is labeled). Go from there and find the fault. Sometimes there is only 1 thing on the circuit, sometimes there is more.
