I've been contending with this problem for a while, but it's been steadily getting worse.

When I engage a turn signal, the signal would come on, and as it tries to turn off, it just dims, and instead of a relay click, I hear something of a very annoying buzz, like the relay click suspended in time. lol. So the turn lights dim but never fully turn off, and get stuck there, which prevents them from coming on again.

I also tried to run hazard lights for a while, and surprisingly, running hazard lights for a while seems to eventually make this problem less severe. The relay still makes somewhat of a sticking noise when the hazard lights cycle, but it gets better the longer the hazard lights stay on.

Am I faced with replacing the relay? And I would have to remove the dashboard to get to it, as my luck would have it?

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84 300SD - bought in April '06 with 237,000 mi

I'm pretty sure your wiper and hazard relay are the same relay, and it's under the center console in front of the gear selector. You might have a bad ground in the signal switch or the relay may be dying or corroded.

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Sure it's under center console? I swear the turn signal clicks come from behind the dash, behind the instrument cluster. Unfortunately, my parts diagrams for this car no longer work on my PC...

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84 300SD - bought in April '06 with 237,000 mi

I had the same symptoms.
On one occasion I found that the relay near the steering wheel had gotten loose from having had the dash cluster out for an unrelated issue. A firm push back in to its socket made the blinkers behave again.
Then recently the blinkers started acting up again and I cleaned that relay contact a few times without any positive result.
I Just carried on with life by manually turning the blinker on and off by hand. Then one day the car would not start and I realized that the alternator was on the way out and I had been driving with a weak charge for a while.
After replacing the alternator and charging the battery my weird blinker behavior went away instantly.
Its worth a check. Put a voltmeter on the battery while it is idling and see if the voltage perhaps is not all it should be but is not quite bad enough yet that you don't realize it.

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84 300SD 274K
38K miles on flatplate heat exchanger and various diesel/veg blends. prior to that 4K miles on unheated veggie blends with kero and DinoD.

That is a good suggestion, and I will follow through, but my alternator has been replaced by a brand new OE unit around 2007 when my original alternator gave out. Still, it's always worth a check.

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84 300SD - bought in April '06 with 237,000 mi

So... how did it work out? any updates?