Thanks, Randy. I'm pretty sure I know what's going on.
You do have the 3.9 engine. Way back in '92 and '93 there were problems with soft distributor drive gears. The 3.9 and 5.2 were both included in the problem, but the 3.9 was over-represented between the two.
Get yourself a Phillips screwdriver and take your distributor cap off. You might need to take the plug wires off of one side (I work from the vehicle left and take that side off) to get the cap up and out of the way. Distributor firing order (clockwise) is 1-6-5-4-3-2.
Once you have a good clear view of the distributor rotor, check it for looseness by trying to rotate it both ways. If you see more than 5/16" of rotor tip travel, the distributor drive gear and block bushing are worn out. Your travel will probably be much greater than 5/16" (8 mm).
This condition and the repair are detailed in Chrysler TSB 18-08-93. Oddly, it says nothing about the driveability aspects, describing it as a noise in the distributor area.
What happened is that the softer distributor drive gear wore heavily in its contact points with the camshaft drive at the rear of the engine. Also, the block bushing that supports the drive gear will be wallowed out, allowing the gear to retreat from the camshaft. Camshafts are never hurt.
All this adds up to a condition where the distributor timing becomes retarded. The rotor tip lags behind where it should be, in relation to the distributor cap. That leads to cross-firing, where the spark jumps to another location that becomes electrically easier... The previous cylinder.
When that happens, you've lost the performance that cylinder normally contributed, plus the air/ fuel mixture is being lit as the exhaust valve is opened by the NEXT cylinder in the firing order. Backfire.
A sensor lives in the distributor, the cam sensor. While it has no effect on ignition timing, it needs to produce its signal within a narrow window of opportunity in relation to the crankshaft position sensor. If they don't fire within that window, the engine becomes out-of-sync and may not start.
If the engine starts, the cam signal is no longer needed to keep the engine running. The odd thing is that an out-of-sync signal plays havoc with the way the engine runs. This is especially true when the engine is cold.
That's because the distributor drive gear does more than just drive the distributor. It also drives the oil pump. Cold engine oil is thicker and harder to pump than later when the engine heats up. During these times, the extra resistance on the worn drive system tensions and stretches all operating clearances to the full-retard position. This is the time when the distributor drive is most-behind in its operation and when it runs the worst.
As a diagnostic aid, you could unplug the cam sensor after starting the engine from cold. If it was acting up just prior to then, it will probably clear up somewhat. Part of the problem is with the cam sync and the other is with cross-firing. Cam sync seems to be the bigger problem at cold engine idle.
You could also rotate the distributor housing a bit to the counter-clockwise position. This would bring both tolerances back into something more like a normal relationship. There is an actual procedure for this, but if you have a worn gear, you'll need to get the gear and bushing replaced anyway and they'll be reset at that time. The procedure is detailed in the TSB.
So just rotate the distributor a bit (if imagining a clock face, make it about 2 to 3 minutes of travel). At that point, most if not all of your driveability should return.
I wish I could tell you the oil light problem would disappear with it, but it seems doubtful. You would need to have your oil pressure checked directly to see what the actual pressure is (could be dangerously low) or you could gamble and just put in a new pressure switch and see what happens.
I'd like to send you the TSB, but I have to confess to being computer-challenged. If you go to the Internet and do a search, I think you could get to a site hosting Chrysler TSBs, if not the Chrysler site itself. TSB 18-08-93.
Let me know if there's anything I can help with, Randy. Good luck with your Dakota!