In my case, GE XL44 gas stove, the problem with it beeping and throwing the F2 F3 code was a loose connection. There is an easy way to check...
I figured it was a loose connection because the temperature jumped up and down when i shook the oven.
To read the temperature:
Put the oven in test mode: Unplug the power for a few seconds, plug the power back in and with and with a short delay, press and hold Cooking Time + Delay Start + number 8. "tESt" should light up on the screen. Now press the Cooking Time button to display the temperature. If it shows steady 900, that means the sensor is disconnected or resistance is too low (F2 code and beeping). If it shows some random number it could also mean a loose connection. Try shacking the oven a little bit, see what happens.
To pinpoint the problem:
Unplug the oven and take the back cover off (the one behind the kepad and screen) also take off the small cover behind the temp sensor (look inside the stove to locate the temp sensor, it a metal rod sticking out of the back wall, in the upper left corner.) Locate the two thin white wires coming out in the back from the sensor and follow them all the way to the top where they enter the control module (the back of the keypad and screen), make note of all the plugs that it passes through.
Now plug the oven back in and go back into the test mode. While displaying the temperature, wiggle the wires and plugs one by one from the sensor up to the brain. If the temperature starts to jump up and down, that's the problematic area. Try unplugging the plugs for the sensor wires and plugging them in.
In my case it was the 5 pin plug that plugged into the brain, I simply pulled out the pins for the related wires and squeezed them so that they make a tighter connection.
If you do have a troubled sensor, you can confirm some of the cases by unplugging the sensor (as close to it as possible, ie. the plug closest to the sensor) and checking the temperature reading, if it changes dramatically (like from a normal 100 to 900 max or the other way around) and stays that way, it's probably the sensor. If it doesn't change, it could be the "brain" - control unit...
If you locate the service manual in PDF format, it describes what the resistance range is supposed to be for the sensor, you can measure it easily with a multimeter at the sensor plug.
Here is the link for the service manual for my stove GE XL44:
http://appliancejunk.com...okingproduct/31-9008.pdfP.S. Disclaimer - if you get electrocuted or brake your stove, or create a gas leak and blow up your city block, that's your fault, use commons sense and caution while working. And if you're not comfortable doing this, call a tech. The sensor wires have really low voltage in them some other stuff in there is wall voltage, so be carefull where you put your fingers...
P.S.S Please let the forum know if this solution worked for you, I'm also curious ;) Feedback is welcome!