My hot light stays on and every thing is cool to touch

If you have the older type stove with the oven knob that has the temperature setting around the outside of the knob, then that part could be bad. It is called a thermostat. That part usually comes with sensor attached. The sensor is the long skinny looking tube that goes through a hole and into the oven.
If you have the newer type electronic stove where you program in the temperature you want, the electronic circuit board could be bad. But the electronic stoves have a over temperature protection built into the circuit board, so that would have engaged, shutting down the stove.
Either way, it is a potential fire hazard and needs to be addressed.

Kill the power to the stove. L2 is always hot to the element. If it welded itself to the frame whats left of the element stays hot. Easy fix but power needs to be OFF

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Tony, this sounds like you may need to have a new burner switch installed, it is called an infinite switch. If it is for the dual burner, make sure you tell the appliance parts store this so they can order you a dual infinite switch.

I have a GE stove which I purchased in 2004, and it is doing the same. I heard beeping coming from the kitchen in the middle of the night . I went to the kitchen and the oven was on by itself. I have been unplugging it when not using it. Maybe this could be a recall.

After letting the stove cool, I pulled the entire unit out from the wall and discovered scorch marks on the wall. Opening the back panel revealed a simple fix: the "element on" light module had shorted and was visibly scorched and melted. Looks like a simple replacement (and some wire stripping where the leads fused). Stove currently works with the affected wires capped off (while I await a replacement module).

glow plug needs to draw 4 amps to open gas valve even if it red hot ...not drawing 4 amps replace glow plug
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no warranty only 12 mo. The switch is bad call dealer - servicer and order part. should be less than $50

The broiler won't turn off even when you press the "Off/cancel" button? It sounds like the broiler control relay is stuck shut. The F3 code means the oven temperature sensor has failed, this probably happened because the oven got too hot when the broiler was stuck on. There is fail-safe thermal fuse that will blow if the oven gets too hot, so the oven won't melt down but you should leave the unit off so you don't blow the fuse.

To fix the relay, you will have to replace the oven control module. I think this is about $200 and not too hard to replace if you are somewhat handy. The temperature sensor is also serviceable, you can find parts at Appliance Parts Pros on the web. Hopefully you haven't thrown the range out yet, let me know if you need further assistance.

Try removeing power too unit for a couple minutes. You can due this by turning off breaker. Sometimes this will reset board. If not, you may need an ERC board.