It sounds to me like one of the shifting adjustments is out of whack with the other one, tilting the basket slightly to the right when you shift.    From what you've described, the right castle nut is just slightly lower than the left one.  Try turning it a quarter turn counter-clockwise and see if your caps align better with your lower cases.  Then fine tune it in a little at a time until you don't notice any tilting of the basket.  Underwoods are N O T O R I O U S about this.

I feel as though I'm about to go down a rabbit hole -- the letters are lined up horizontally, upper case and lower, but definitely when I shift I notice the right side of the basket tilting down farther than the left side. So that would indicate that I need to turn the right side castle nut clockwise a fair amount, so as to raise the stop higher and (hopefully) bring the upper case and lower case into vertical alignment. At that point, I will need to turn both castle nuts in synch with each other until I get the upper case and lower case back into horizontal alignment. Make sense?

Oh -- actually when I was talking about turning the two nuts in synch, I meant turning them both counterclockwise so as to bring the upper and lower case back into horizontal alignment. (Turning the right hand castle nut clockwise so as to try and correct the vertical alignment has thrown the horizontal way out of adjustment, but has made some slight improvement on the vertical alignment.)

But you make a good point -- I should turn the right side nut clockwise, and left side counterclockwise, to hopefully get things truly vertically aligned, now that I seem to have the upper case stops pretty much lined up evenly on both sides.

Then, at that point, I believe I should be turning both nuts counterclockwise, in synch, to bring things back into horizontal alignment. Sound right?

Many thanks for your advice here!

Well, I've got it a little bit closer, I think. But I found that the bolt holding the left side castle nut actually started backing out of the plate into which it's threaded, so I am not sure how to tighten that down. Also, I notice that when I depress the shift key, the entire right side of the basket drops a little lower than the left side -- not that the stop on the right side is lower. So I am wondering if there is something more basically out of whack, which can't be compensated for by adjusting the castle nuts. Maybe the connections between the shift keys and the basket? I may fiddle some more with tightening the right side nut and loosening the left side nut, but I am not optimistic that it'll make much difference. but thanks so much for your invaluable guidance here!

You had mentioned that jamnut before -- not really sure what I should be looking for, but as I recall (would need to take another look), there were no jamnuts, but rather screw-slotted nuts. And none of them is really accessible.