Well, I couldn't find one with limited slip, didn't have the time or space to swap it out anyway, so I took it to Hooper's Rear Ends in Sun Valley, Ca.
He quoted me a price of between $500 (if only bearings and seals), to $700 if he needed to swap in a set of used gears. He ended up swapping in the gears, but I am pleased overall.
Very nice people. It's apparently been in business for a very long time. Their office is filled with hundreds of gears, axles, differentials, and related parts. It took them about a day and a half, and I am really pleased.
They were puzzled by the fact that the rear end wasn't howling, or groaning, or whining, even though the pinion was flopping around in the housing.
Thanks again for all the good suggestions and info.