Chris Nelson wrote:
So I picked up a used tandem (Trek T900) and its a hoot but it has
some rear shifting problems. More specifically, when shifting the
chain momentarily sits on top of the cogs before settling in, creating
a temporary cadence increase. Once shifted, it runs fine. It happens
on up and downshifts, and I believe to have it adjusted correctly as
far as indexing, because it shifts flawlessly in the workstand. It
only has problems under load.

It has very few miles on it as the previous owner only used it a
couple of times, so I believe it has nothing to do with wear. It has a
mixed set of drivetrain components, which may have something to do
with it.

SRAM 850 11-32 8 Speed Cog
SRAM MRX 8-Speed Twist Shift
Shimano 8 Speed Chain - IG
Shimano Deore 9-speed derailleur

I've lube and cleaned it a bazillion times and I ready to start
replacing stuff in order to get it to shift better, but I'm unsure of
the order that I should replace stuff and what to replace it to.

Sounds like simple cable tension. Turn the barrel adjuster counter
clockwise a bit?