I have an Amana First Edition stove probably 10+ years old. The front surface burner won't turn off. I cut the breaker and it turns off, but when I turn it on again it won't turn off.

SOURCE: GE Spectra range

All of the items you mention that do work, are 120 volt items.
All the items that do NOT work are 220 volt items.
Sounds like you have a voltage problem, you will need to use a meter to check if you have 220 volts a/c to the outlet the stove plugs into. Take care when doing the following, you will be around live voltage and could get hurt. If you are uncomfortable working with a live circuit best to call someone in to do it for you. Check for 220v at the outlet for the stove (where the stove plugs into), if you have it there, go to where the cord attaches to the stove and check at that point. You should have either a 3 or 4 wire cord on the stove, On the rear of the stove there is a terminal block that the cord is fastened to. Between RED and BLACK you should have 220v, between RED and WHITE and BLACK and WHITE you should see 120v. Another thing to check is that the nuts holding the wires from the cord to the terminal block on the stove are turned down tight. A loose connection there will give you the same problem you have as well.

SOURCE: Burner will not turn off

I would first check the infinite control switch (#WB24T10029) the part the knob slips onto. The next thing I'd check is the erc (#WB27T10231) control board. Hope that is helpful!

SOURCE: I have a new ge range. When I turn on the left

mvek,
If you could see through the glass, you'd see "temperature sensors" located near the elements. They're very sensitive little guys, and rightly so! Their job is to detect temps and energize the warning light on the cooktop so users won't get burned.

To answer your question... It appears that there is an issue with the front left sensor (and possibly the right as well). The temp warning lamps should NEVER go off immediately. Please call for warranty repair (1-800-GE-CARES). This is an unsafe condition for you and your Family. The warning indicators are there for a reason and should indicate high temp's as long as high temp's are present.

SG

SOURCE: I have a an AMANA gas stove burners work but oven doesn't

First check the book to see if you have a igniter on this stove if so it takes a moment or two to get started, if not you need to light pilot
make sure gas is getting to burner when called for by the thermostat or oven control..

SOURCE: Hotpoint Glasstop Stove - Burner turns on by itself

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.