EMERGENCY AND TURN SIGNAL FLASHER-- NEW RELAY DOESN'T WORK?

The description sounds like either a failed multi-function switch hazard assembly or a bad connector on it. That is the only part that those all share. The turn signal uses a different flasher than the hazards but the turn and brake lamps all route through the switch, but on different circuits. Does the switch feel loose or crunchy as you move it? Try this, pop the hazard switch a couple times like on-off-on-off rapidly, I have seen the contacts fail internally and cause a lot of strange light issues. The metal contacts would corrode and melt the internals of the hazard switch, then you had a brake light on one side, or the hazards would blink in the front but not the rear. Usually hitting the switch would tell you it was the problem because suddenly things started working a bit or the failure pattern would change when the switch was cycled. On your car the switch is a bit more complex but they still fail the same way.

That switch isn't fun to replace as it's buried in the steering column. To get to it you need to pull the steering wheel, then the lock plate and then pull the lever out of the switch and remove the screws and clips on the switch, then the switch is disconnected under the dash and you fish the wiring up through the column. Then reverse the process to install the replacement.
The part number is either GM 1995967 or GM 1995946, the difference is if your vehicle has "accessory control" through the switch. AKA radio controls on the steering wheel.

Agree with you on the folks not understanding hand signals, they all seem to understand one hand sign and the horn and use both in anger.

Thanks, Steve. I nod to your experience, of course, but it seems even the flasher might account for at least some of the variation in front and rear, left and right, already observed.
Yes, I have only the Novita EL12 as a sample of one, so I cannot speculate other than to hope to find another flasher that shows a different lighting pattern-- even a flasher that restores all functions. In that forlorn hope, I plan to contact the OEM flasher source for my after-sale options, so I can return the flasher if it does not work.
In any case, you are probably right about the ground fault and/or contact problem.

Unless someone rewired the car all the flasher modules do is use a thermal strip or electronic timer to open and close the circuit that feeds power to the selector switch. It cannot change which lights actually flash when selected. Plus you have issues with both the hazards and the turn signals, those use two different flashers and power circuits. The common part is the switch inside the steering column.

Earlier, you had asked how the turn selector lever felt when I pulled it-- I found no unusual resistance, or crunchiness, but only the normal smooth movement into right or left position resting "notch".
Today, I bought a steering wheel puller.
(Thanks for the exploded diagrams-- looks intimidating, but if it were easy....)
Will keep you updated on what I find.

Thank you, have a good weekend.