Without being there, it's tough to make a call. This much is true, NON-console, column shift 78-88 A/G Oldsmobiles (and probably others) with 3 speed auto trans all used the same transmission link at the transmission. GM P/N 10021792.
It does sound like an adjustment issue from what you describe. First, I'd get the car safe and stable where you can get under it (jackstands, lift, etc.) then put it in neutral at the column. Then I would check to see the linkage is all intact and there (check bushings too as if the column bushing is worn out, you'll have major linkage slop). Replace the bushings if needed. Make sure the transmission lever at the transmission is in the neutral position. If not, loosen the adjusting bolt near the end of the down rod from the column. Don't remove it, just loosen it. Then put the transmission lever into neutral by hand. Tighten down rod bolt. Now place the transmission in park and see if it goes there. Select all the gears and see how the transmission lever aligns. If it's still off, then something else is the problem.
With a column shift and 83/84 H/Os, there's also an indicator on the instrument panel to show you which gear you should be in. This has no brain or anything to do with any of it, it's just an indicator. If you find you're in neutral and it's showing "R", then you need to adjust the indicator. If everything is still there and working, there's a friction clip on the bottom of the shifter bowl (take the vent panel off under the dash below the steering column to access it) that can be adjusted one way or another to show you the gear you should be in.
If that all works and you're satisfied with it, then check your reverse light switch on the lower top of the steering column. If it needs adjusting so the reverse lights come on in reverse, then do that.
There is no "neutral safety switch" on G-bodies. It's built into the column. This is why it's so important to get things adjusted right. Because if the COLUMN thinks it's in Park or Neutral, it doesn't matter what the transmission is doing, so it may or may not start when in gear.
I'll post a couple pics in a minute...