Probably one could and should break out Bentleys and examine wiring diagrams till you can conjure up a scenario that would give your exact experience. I admit, I'm not up to it.

But I'll make a couple somewhat educated guesses.

First, I wonder if the voltage regulator has gone nuts-o, allowing random over and under voltage. This would give the batt/alt warning light when low, and maybe give you a loud turn signal relay click when high. Would also drive the ignition/injecton system nuts; I'm sure it's protected against wrong voltage so it would likely shut down when voltage gets outside the allowed range.

If you like, you can rig a voltmeter to the battery or alt and run the lead into the passenger compartment. I would do it. Then you can observe voltage while driving. Don't rely only on a 12V source from the fuse panel; you want a reliable reading from the source. You can ground the voltmeter to the tab at rear of auto tranny shift lever housing, that's a good ground for an alligator clip.

Regulator is installed to back of alternator. You can change it without removing the alternator, but do remove the battery ground first.

Second, maybe the relay flasher is bad. They're pretty easy to replace. To access it, remove the plastic panel to the right of the driver's right foot (USA cars). It's clipped to the vertical frame piece at the corner of the center console. Sometimes comes unclipped and falls under edge of carpet. Has only three leads going to it.
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Sven: '89 245 NA, 951 ECU, open-front airbox, E-fan, 205/65-15's, IPD sways, E-Codes, amber front corner reflectors. Wifemobile '89 245 NA stock. 90 244 NA spare, runs.