About 2 years ago, my car had the same exact problem as you described. From a standstill, I would hear a singing "wobbling/chirping" noise. The harder I stepped on the gas, the noiser it would become. It would go away once I hit 30 or 40mph.

The culprit turned out to be the center support bearing.

My independent Volvo mechanic changed the bearing and my rear pinion seal for around $80. Ever since the bearing was replaced, the noise has ceased.

I would have a reputable mechanic inspect the U-joints. If they seem okay, have him replace the center support bearing and you should be good.

Who quoted you "up to $2,000" to have it fixed? Midas?
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Regards,
Eric Staufer,
'89 244DL 118k
My 240 Page

i would agree that it's probably not the rearend, as they're considered to be very rugged units. used units can be found for $100 if not less, and can be installed in a few hours.
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87 245 V8, 88 760TI, 93 benz 190 2.6

Sure could be the driveshaft center support bearing. Usually this tends to "sing" when cold, but quiets down when it or the weather are warmer. It's relatively easy to drop the driveshaft and cenetr bearing and spin it to listen for bearing noise.

The Volvo differentials are about as sturdy as the Rock of Gibralter.

If you find you actually do need a rear end, shop the boneyards for a low-mileage used one of the same ratio as yours and with the electronic speedo sensor. Because they almost never fail, there should be plenty available for pocket change.

It's unlikely your diff is dead.
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Don Foster (near Cape Cod, MA)

u joint noise, chances are not the dif. Put it up on a lift and inspect them. Also check the center bearing and support