ATMega328P - faulty ADC behavoir - update
First of all make sure that your code is as posted and there is no funny business other than what's posted.

A dc results can change from a variety of reasons. I would put a probe on a3 , vcc / avcc, and aref to make sure that they are not going up and down.

After that, tie a2 and a3 together and plot them out on the same chart and see if there is a pattern.

The key is to make sure that those implicit assumptions you made are indeed true.