ADC noise measurements gives bimodal (double hump) distribution in output values
What kind of voltage source is your resistor connected to?
My first guess would be that the source your resistor to has some square wave signal present in it.

Are you properly decoupling your power supply? Is the power supply the same in both of the two experiments you described?

Perhaps your instrumentation amplifier does not have good power supply noise rejection.

What is the voltage reference of your ADC?

Do you have access to an oscilloscope?
It would be interesting to look at the output of your instrumentation op amp to see if there is any noise in the signal.

Have you tried changing the voltage level (to something other than half-supply)? That shouldn't change your results - but if it did that would be interesting and it would be very useful information.

I understand you are not going to post any schematics; I strongly disagree with the decision.

Edit:

Turn up the gain of your instrumentation amplifier. If you see a wider distribution, this could imply the bimodal signal is entering your measurement before or at your instrumentation amp. If you don't see a change in the distribution, you may conclude that the error is entering your measurement at or after the amplifier.