Now you have me wondering if the problem was just the result of an autoranging multimeter changing ranges suddenly.

 

There are various types of pots that are not linear. I haven't delved in this for quite a long time but there are also "decibel" gain type pots. Just get some new pots.

If you are really interested then plot a graph by taking readings with percentage of turn of the knob in 10% increments plotted on the X axis and the resistance readings plotted on the Y axis starting at 0 Ωs at 0% turn and also invert your results. Plug in the readings into various electronic mathematical formulae and you will see which mathematical relationship the pot is made for. Remember that the result may be scalar so you might have to do a bit of number crunching and custom tailor your results to a formula that will fit your graph curve, but it will satisfy the correct relationship/function curve results.

You may be able to recognize the curve through familiarization of different math functions.

Danny