Switching a current source through a MUX
You probably need to short the unused current sources to ground with another set of analog switches.

When a current source is not connected its voltage will rise to the maximum level and it sounds like it is then high enough to get through the multiplexer.

4053 2:1 multiplexers could be used with the input current source to the common terminal and all of the NO outputs tied together as the multiplexed output.

Alternatively you could use diodes instead of the multiplexer and short all but one current source to ground with analog switches.

This will take more devices than using just a 4051 multiplexer.