Contactless electrical connection
An electrical connection is, by definition, not really contactless. For measuring resistance you need (continuous) current to flow, and for that you need a connection. You can work with capacitive or inductive coupling, but that needs AC.

For your problem you can try to use TI inductance-to-digital converter, e.g. the LDC1000 (there is also a not-so-expensive EVM available). With that you can mount a small coil on your carriage, and just by the changes in its relation to the shaft the measured inductance changes. And since it has 24 bit resolution this might be sensitive enough (depending on your resolution requirements).