Thanks for all the suggestions - the tax man has interfered with me responding in a meaningful way before now.
Phil and others suggested a good bike fitting, and that is something I had planned, and forgotten, for after I got set up with clipless pedals (which I have been using for a month or so now). I have some good references for a local guy to try.
I did see the "riding position discovery" thread when it first started, but now I see there is MUCH more information in there. I've read through it once, but I think it will take several more reads before I fully comprehend it.
I went on a 50-mile charity ride with my niece and there was not much of a problem until about 40 miles, and then only on the right side for some reason. I have been standing up every few miles, weight off the saddle, previously, but made even more effort to do so on this ride. I think that helped. I'm not riding at max speed since I'm with my niece, so I could probably get more weight on the pedals riding alone. I didn't have much soreness afterwards, but I definitely would not have been able to do many more miles, and I'd really like to build up to full centuries by the end of this season.
There is only one hard chair in the house that makes me notice; the thing that really made me notice the sore sit bones was when I was at a party and ended up sitting on the stone fireplace slab for half an hour.
I attached a picture to give somewhat of an idea of my "typical" position. I flipped the stem when I first got the bike, because the lower position was making my hands/wrists numb almost immediately. If I understand the "ride position discovery" thread, raising the bar may have been the wrong thing to do. However, when I started, I had very little core strength - couldn't even do 10 situps. Now I can do 40 (urged on by my niece, who can do 130), so again if I understand that thread, lowering the bars could actually reduce the weight on my hands by causing my core to hold up the upper body instead.
So, first to get a bike fit - I am pretty happy with my saddle height, but I think the forward/back and bars need some adjustment to get a more efficient position. Then I'll start experimenting with the "ride position" ideas and finally work on the saddle itself when my position is "finalized".