Problem powering ESP8266 with two 1.5V AA cells
Modules like the ESP8266 which have no on-board regulator but do need a regulated power supply. Indeed, a power supply voltage that is too low will cause the module to reset.

What I would do in your case is get a boost module like this one to convert the voltage from the batteries to 3.3 V. On the module there's a trimmer to set the output to 3.3 V, set this BEFORE connecting the ESP8266 because such an upconverter module can easily generate 12 V which would fry the ESP module !