the ATmega sporadically resets as if it is drawing too much power.
That is not the only cause of random resets.You have to test what the power supply is giving you using a meter but better would be an oscilloscope because that would show the dips and if you had too much ripple.
The problem is that when I plug in the ESP wifi module, the ATmega sporadically resets as if it is drawing too much power.
Try changing C3 to 1000uf.Also add .1uf close to the regulator input to ground..
0.65A from a supply with a maximum output of 0.85 does not really count as the supply being "far in excessof what is necessary". Allowing for inrush current it seems adequate though.You would do much better to derive the WiFi supply from the 12V with a DC-DC converter, not a chainof linear regulators, as that is much more efficient.How hot is the 7805 getting? Perhaps its shutting down from overtemperature?
you have a multimeter, have you measured what the power consumption is of the various parts ?