Whew!

I really appreciate all the help. After reading everything - Angie's List sounds like the best idea! I'd tackle it if I had Bobert or Allan right there to check, inspect and instruct every step of the way. I have an unfortunate track record with attempting all but the very simpliest home repair projects.

From what you have posted, and after poking around a little more I realize the basket drains were not installed "according to Hoyle" to begin with, and therein lies the problem.

1. Instead of using plumbers putty between the sink and the lip of the drain baskets, they used silicon gel. This means the drain baskets have always slipped and slid around a tiny bit when pressure is applied (such as scrubbing the sink, or pushing the stopper basket down to close the drain.)

2. The nuts under the sink turn very easily. (The first thing I tried was to tighten them.) They will not snug up tightly to the sink. One of them is so loose that while I can carefully screw it up so that it will stay, with a good jostle, it drops back down again.. John may be right, the threads may be stripped, at least on the nuts, which appear to be a galvanized metal. (and they do have those little nubs at intervals around the perimeter.) There is residue of some sort that may be plumbers putty, but is not silicone, and no huge amount of it, whatever it is.

The nuts were either fractionally too big or the threads got stripped during installation. I'm guessing the plumber or handyman - I can't remember now who installed the sink, just that it wasn't the thief who left the stalactites of silicone dripping from under the old sink - compensated for the nuts not snugging up by applying a sealant. It worked for awhile, but because the baskets were not clamped tightly the seal between the sink and the drain baskets eventually broke.

I'm not brave enough to do it on my own without on-location guidance.

Please don't think you have wasted your time. You've provided enough guidance for me to realize this is not a diy project for me.

The side of the sink that drains straight down into the trap developed this problem, on a minor scale, several months ago. A bucket works there fine. I had been putting off having it fixed because I also have problems in the bathroom that have weakened the floor and rendered the shower (but not the tub) unusable. (a legacy problem that appears to have been half-assed repaired many times over the years by the previous owners, including a half-assed repair so that it passed the pre-purchase inspection.) To fix that problem and fix it right is going to cost several thousand dollars. I've been saving up for that repair, and thought to have the sink dealt with at the same time, since plumbers would already be here. Last weekend, however, the other side of the sink developed the same problem, only worse. Since the drainpipe for that side of the sink runs horizontally to join the pipe lined up with the trap, the water runs all along the exterior of that pipe. There is no bucket with a wide enough diameter to collect it all.

I have terrible karma when it comes to finding good and reputable trades and repair people. I think I must have "sucker" written across my forehead.