Hi. I just got another bike, semi-new Schwinn, apparently hardly used for years.
I spent some time this morning making little adjustments and short trial runs.
I found that, when in the two highest rear gears, the chain would seem to bunch and lurch.
Of course, I couldn't see perfectly what was going on.
The teeth are fine on the gears,
so I thought maybe the derailleur wasn't picking up all the slack in the chain.
I adjusted it out all the way, trying to make it tighter.
It seemed to help a little for the 2nd-highest gear, but I may have been imagining it.
So, for now, I just adjusted the tracking max to not allow it into the highest gear.

The chain doesn't seem to spring much on the jockey wheels,
but I've gotten used to using a fairly short chain on my Peugot, so I'm not sure.

Does anybody have an idea of what might actually be going on?

What you describe sounds exactly like the problem I have on an 11 tooth small cog. If I switch cassettes to one with a 12 tooth small cog, no problem. So I left the cassette with the 12 tooth on. I couldn't figure it out either. This was on a 9 speed system. The problem was with a SRAM cassette, the Shimano cassette with the 12 tooth small cog does not have the problem.

Your chain may be significantly too long. Shift carefully into the big-big combination (largest chainring and largest cog). If there is still a lot of room for the lower derailleur pulley to move forward, you can shorten the chain until it just allows the shift into big-big.

By high gears do you mean the two smallest? In the rear the smallest are the highest.
This problem would be typical for cogs that have been worn due to riding with a chain that has stretched too much causing uneven wear on the cog teeth. When replaced with a new chain the chain links do not mesh with the worn cogs. If that is the case you will need to replace the cassette or freewheel, whichever you have. You didn't give us a lot to go on. Have you measured the chain for stretch?

edit: Hillrider posted at the same time so I had not seen his post. I agree with his assessment.

This may be a different problem. When using an 11 you must also use the special lockring for the 11. The chain will ride the lockring instead of engaging the cog properly if the normal lockring is used with an 11.

Yes.

Yeah, it seems to me like it's due to too much slack in the chain.
Obviously, I can't get any more tension than I've eked out already, so it falls to shortening the chain.

Thanks, guys.

I think I may need to check chain length in my situation. That didn't occur to me. In my case, both cassettes and chain were new, and I used the lockring that came with with each cassette.