Going on nothing but the intermittent shut offs with immediate re-start, no codes being thrown, and the otherwise unexplained odd behavior of the cruise control, my suspicions would be with the cam position sensor or its wiring (or timing that is right on the edge of some error threshold).
If you disconnect the cam position sensor, does it run without cutting off? It will be limited in rpm without the cam sensor but otherwise should run normally if the cam sensor circuit is causing your problem when connected. (The ign. runs off the crank position sensor up to IIRC 2500 rpm and then runs off the cam sensor above that RPM.) If doing that stops the shut offs and odd cruise control behavior, at least you will narrow down the possible cause. If the shut offs continue with it disconnected, you will have lost nothing other than your time of running that experiment and will likely eliminate the cam sensor circuit from being the problem. I honestly donít know if cruise control works with cam sensor disconnected, but if it changes from current strange operation to total inop., then that tells you what you need to know either way. The c.c. may operate normally with the sensor disconnected, which would mean that it is flaky cam sensor function (vs. complete absence of cam sensor) that affects c.c.
Community experience is that cam sensor issues more often than not result in intermittent shutdowns with no codes. The theory is that the PCM has to see operation with Ďxí counts of bad signal to set a code, but that if it fails just hard enough to stop the engine before reaching that count, then no code (and apparently that recycling the key resets that bad sensor signal counter). It also was learned over time that c.c. not turning on normally, or turning on and unexplainably turning itself off, is often a precursor to hard cam sensor failure. Hence my theory that your symptoms point to, but donít necessarily prove that, there may be a cam sensor or cam sensor circuit issue.
Consider also the possibilities of timing being right on the edge of being off enough to prevent running, or that you may have gotten a bad OEM sensor (it happens; could also be crank sensor, but past forum history says always or almost always itís the cam position sensor). May also be incompletely mated connector giving intermittent wire connection. Demate and re-mate the two rectangular connectors right next to the brake booster, also other PCM and engine connectors. Make sure all are fully mated.
You most likely have removed battery power for several minutes at a time for one reason or another, but if not, wouldnít hurt to do that in case resetting computers will help.
'98 LXi - Later Concorde gages (black w/ chrome rings)/'99 LX - LHS gages (white) - HIR bulbs