That ugly word rears its head again ! What kind of breadboard ? How is the circuit laid out ? Power supply connection do look ugly, but we need to know that you have everything bypassed, and in the right places too. Grounds around the ADC are particularly critical for lowering noise.

Its such a simple circuit, I think a basic PCB could be done very easily.

May I say what a lovely job you have done in presenting the problem to us. Perfect !

Steve