exoro:
Before I purchase a set of 10ks, is there any way I can get the 100ks to work?
Use 100n caps from each mux input to ground, one for each pot (no cap at the output of the mux).
When the mux switches to the relevant input, the Arduino sees the low impedance of the cap instead of the high impedance of the pot.
Now you can use any pot value, even >1Megohm.
Reading the analogue input twice (or more), and discarding the first read is always wise.
Maybe also wise to ground any unused mux inputs, and not select them in your code.
Leo..