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You almost certainly need a replacement gas valve or a combustion safety valve. Your issue sounds like the combustion safety valve is tripping when the heater gets hot. Alternatively, there could be an issue with the gas control valve. In my experience, the gas control valve is open the culprit after the thermocouple. Your unit is electric start. The gas control valve is easy to replace and costs about $165 on SupplyHouse.com
BTW - I am not even sure your unit has a combustion safety valve. GE says it does, but I don't see it anywhere on the parts breakdown. Possibly they just print that in all the manuals.
You can probably do this job yourself. It is very easy and doesn't take more than 20 minutes. When you get the part, we can have a video chat ($25) and I can walk you through it.