The gear selector should always be able to go into Park without any other aid. Something is blocking it, possibly something down in the bottom of the gear selector housing at the back (less likely, but worth checking with a flashlight through the nylon fibres) or in the forward upper part of the gear selector housing like a broken microswitch. There should be something in the 700-900 FAQ on dealing with that microswitch problem. It was a plague for 960 owners.

The gear selector will only come out of Park if the key is in the ignition in the KPII position (ignition on) AND either your foot is on the brake pedal OR you depress the shift override button at the rear of the shifter. In either case, it operates a solenoid to let the selector go out of Park. If that solenoid was somehow jammed in the out position then that could be the source of your shift lever problems. And, as was pointed out, you should not be able to remove the ignition key unless the shifter is in Park (and the steering wheel isn't hard over to one side binding the locking mechanism).

When using the overdrive button on the back side of the gear selector, the arrow in the instrument panel should immediately light (or go out). If not then either the overdrive relay is faulty (or loose) or the overdrive switch wiring that comes out the bottom of the shift lever is broken or the overdrive button is dirty. On 240's the overdrive relay was often the culprit for this particular type of problem. I haven't heard of this sort of thing being an issue on the 900's, but anything is possible. Technically this feature is called overdrive lockout which prevents an upshift into 4th or forces a downshift from 4th to 3rd. If the arrow doesn't operate in synch with the button then you need not look further than those problem areas. BTW I don't know why they moved the overdrive button to the back of the shifter. I liked it just fine where it was. I even didn't mind it when it was a stalk on the steering column on the Amazons. Probably some design effort at a universal shift lever for left and right hand drive markets, like they didn't have the technology to keep using the 700 levers, but I digress.

When you are at full cruising speed (like above 30 mph), the transmission should change relatively quickly out of 4th down to 3rd (or back up from 3rd to 4th) in synch with the arrow on the instrument panel. If not then either the overdrive relay is faulty or the overdrive solenoid on the transmission is faulty (sticky). In your case it could be either as you are eventually able to get overdrive lockout to work after repeated use of the overdrive button. Perhaps the easiest thing to do is try tapping (as in beating) on the relay while the arrow is showing and the trans won't drop down from 4th (or vice versa, the arrow is not showing and the trans won't upshift out of 3rd). If tapping (beating) on the relay helps then you know that's where the problem is, otherwise it may well be at the transmission solenoid (wiring or a sticky solenoid). Check the 700-900 FAQ here for more info on solenoid problems. If the trans oil looks a deep burnt brown (as in it hasn't had a recent flush) then a thorough fluid flush can sometimes help with a sticky solenoid -again, see the FAQ.

BTW, your 1993 940 car should have an AW71 transmission, at least the North American market 940's. The 1994-1995 940's got the AW70L transmission with the lock-up torque converter (almost like a 5th gear). Be happy nonetheless, the Aisin Warner transmissions are pretty bullet proof.
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Dave -940's, prev 740/240/140/120 You'd think I'd have learned by now