If the sensor was bad, showing a signal stuck more in a high or low limit but not totally shot, it would run rough and not show a code as the computer see's it as working........in essence it is working, but not in normal parameters....i.e.; high to low, instead of high or low. If it's totally shot (or unplugged)......meaning no signal, the computer will show a code. I would surmise that maybe a sensor showing some code at all might run less rough than one that is totally shot, but that's completely an assumption. In either situation, code or no code, it should run rough enough to easily notice. In both situations, the sensors are bad.......just one showing a code and the other not.
As far as the computer running off of an old signal if the sensor was totally shot, no......it does a check of all systems on startup, which is why it throws codes when something is totally bad. If the CPU doesn't receive the proper signal from a sensor, it will throw a code and switch to default, "open loop", or LHM (limp home mode). Sensors that are totally gone pop up right away (no signal).....sensors that are borderline either won't, or might after a few miles. An engine should start, and run rough, but like I said, it's possible the 3.8's have a CPU function that doesn't allow it. I can't see why GM would change that process, but it doesn't mean they haven't.....so I conceded there.
When I bought my 00' GT it didn't run at all (which is why I got it for what I did

.......but I knew it was okay mechanically. It wouldn't start at all but failed to show any crank sensor code. Even when we read the separate circuits with a scanner (not a reader), it would show a signal (the tach signal/wave only), so, not knowing it was a dual wave, we figured it was good. What we failed to realize was that it was a dual wave sensor (not a single)......tach on one side, signal on the other. We had tach, but no signal to start. I planned on replacing the electronics anyway with new stuff I had back at the shop, so I tossed on a new ignition module and a new cam sensor first......nothing. We dug deeper and found that the crank sensor operated with that 2 wave system so we replaced it and bingo.....fired right up.
I think Kev's problem is maybe two fold.........he replaced the crank sensor (assumably a new one), and it ran, but misfired, so.......assuming that there are no codes.....possibly a bad ignition module ?, bad coil(s)?.......TPS even ?......hard to tell. Best way in my opinion is a scanner.......it'll read everything, including the individual parameters of the sensors (i.e.; high to low, etc.)...........
I think even a horribly plugged converter could simulate it to a degree.......which is what I'm finding now. Anyway......hope he figures it out.