So if I experience the same issue with a known good battery, then my problem lies further in, correct?
Yes that would be correct.
The next step is to check for voltage drops.
1: Between the earth terminal of the battery and the actual engine block.
2: Between the Positive battery post and the main starter motor cable terminal.
Very quick and simple to do, no good getting another starter and finding you got the same issue.
To do this you will need a multimeter, or voltmeter. Ill find a pic and post how to. Got some here somewhere from electrical course notes I wrote yonks ago.
This pic shows checking voltage drop between a battery post and terminal, you'll notice one probe is on the post and the other is on the terminal. We set the meter on the DC volt scale, say 5 volt scale if you can. If the meter is autoranging then just set it to DC volts. The engine is cranked and the reading checked. It should NEVER be more than .3 (that's point three) of a volt.
Let's check point 1 above, for voltage drop between the
NEGATIVE BATTERY POST and
ENGINE. Hold 1 probe on the NEGATIVE battery terminal and the other probe on a good clean EARTH point on the ENGINE BLOCK or HEAD. Meter on DC volts, crank the engine for say 5 seconds and note the reading. If the reading is more than .3 volt you have found a potential cause of your problem. If the reading is above .3V then test each connection point individually
Engine block to earth cable connector
Negative battery terminal to negative battery cable terminal
If they check out ok, less than .3V but you have more than .3V overall you have a dicky cable.
It is also a good time to repeat the same test from the battery negative post to the vehicle frame/body.
For point 2 above, to check for voltage drop on the positive side of things, put 1 probe on the battery positive post and the other on the starter motor main cable connector not the battery cable connector but the actual threaded starter terminal. Again crank over and ensure the reading is not more than .3 of a volt. If so work your way back as in the previous earth test.
Another great use for a multimeter is to check the battery voltage drop whilst cranking. With the meter on DC volts scale say 20 Volts, or again if autoranging leave it on DC Volts. One probe on the negative battery post and the other on the negative post crank the old girl over for say 10 seconds. If the voltage drops below 9 volts forget it. I don't know the specs or anything about Oldsmobiles being Aussie, but Id be willing to wager, and I would win, if the battery voltage dropped below about 8 to 8.5 volts the computer (for want of a better word) would go out to lunch. You would need the battery to hold at least 10 volts for a good 15 seconds to say you have a good battery.
Having said that of course a tired old starter CAN pull battery voltage down pretty quick but its not that common. A bad starter motor however will NOT cause voltage drops at any points mentioned earlier.
Very difficult to diagnose these things from a distance, but a good auto electrician would definitely check the above as a first point of reference.
Good luck, and I hope you can understand my scribble, if not holler.