Noise problem on input of an analog to digital converter
The main problem is that the voltage reference (LT1019) is not suited for that ADC. The ADC requires substantial decoupling on the REF inputs because it is an unbuffered input and will glitch merrily at whatever sampling frequency you are running at. However, the LT1019 likes to run without output decouplers. It's a very old device and the data sheet is poor and just does not tell you how good the LT1019 is at dealing with high frequency switching noise caused by the DAC. More modern devices show a graph of output impedance versus frequency to give you better facts to work with.

This is also true of the inputs but at least you are using a buffer amp (MCP6231). However, that buffer amp has a very poor equivalent input noise (50 nV per root(hertz)) and, now that I've checked, does not specify what output impedance it has (versus frequency) either. Poor choice of op-amp and poor voltage reference.

There may be other things wrong too but these stick out to me as bad design choices.