Gentlemen, thanks to all who have been patiently advising me on my S3 carbs. The bike runs very well now. Let's review what I learned to get here.
Cracked floats replaced by a used set from usernameandpassword - Thanks Richard, they worked fine.
Bike still terrible in the mid-range but not drooling gasoline all over the place.
There are two more things that I had to do to eliminate my problem. Through many tests I determined that the RH cylinder was the culprit. Just to be sure I was looking at a carb problem I switched the center and RH carbs. OK - the problem moved to the center cylinder. It loads up sloppy rich and fouls out quickly plus the issue with running ragged in the mid-range.
As I said, there were two things that I had to do to finally get this to work. The first one I am embarrassed to admit but I will for the sake of completeness. The needles were installed incorrectly. I put them in the same way that they were in the slides when I bought the bike. They were just slipped in from the top so that the retainer ring rested on top of the plate at the bottom. As I thought about how the slides work I realized that the needle could easily hang up after the slide moved down. That would cause some real confusion in the carbs. I checked the exploded diagram in the Mikuni manual and as I suspected, the needle goes in first, then the plate. This does two things - it makes the needle follow the slide up AND down & it dropped the needle position just a bit which I needed.
Some progress at this point. With carbs back in original positions, LH and Center are working pretty well with good plug color but RH is sloppy rich and fouling out the plug as before. Not bad since this is the first time I have had good consistent results on two cylinders.
The next thing I did is a bit weird but the more I thought about it, the more I believed it would fix my problem. I had nothing to lose because at this point I was shopping for replacement carb(s). On another thread, long ago, I posted the following picture and asked if I should worry about the missing plug at the bottom of a tube in one of my three carbs.

See the one on the left has an open tube and the one on the right has a plug with a tiny orifice in it.
I was told that it has nothing to do with my mid-range problem and that it was part of the enrichment system. I said, Duh, you're right. It looked like the worst case is that it would deliver too much fuel when using the choke on that one cylinder. I continued my build without thinking much about it again.
After a few more failed efforts to get the bike running right, I took a few days off and just thought about it some more. I got it down to the problem being in just that one carb. The only difference that I can see is the missing plug. How can that affect the operation of the carb? I think that all is well at idle. It will idle all day long. When I start accelerating that cylinder gets all sloppy and the bike runs like crap unless I run it at full throttle all the time. It does not like partial throttle operation. I now suspect that as engine speed increases, inlet pressure drops enough to start sucking fuel through the enrichment valve. The enrichment valve is supposed to block the flow but without the orifice in the tube there is atmospheric pressure pushing fuel out of the bowl and flooding the enrichment valve. Enough fuel gets by to cause a super rich condition. With wide open throttle the condition becomes less problematic and the engine runs OK.
So I plugged the bottom of the tube with a piece of stainless rod that I silver soldered in place.

There is no orifice drilled through, but I just wanted to get rid of the big hole in the tube.
I did all the normal carb adjustments and then took it for a ride. It runs great.
I will go find that other post on the question about the tube and put this information there so others might not fall in this trap.
Time to put some miles on the S3.