This is tough one to understand. I've mulling it over all day and I just can't see how both valves could be left open to release gas without 2 failed components every time that also recovered on their own. It's possible that you have a failed gas valve with the secondary valve stuck open but that's pretty rare. If that was the case there should be gas flowing as the ignitor heats up as well. At that point the booster coil is energized so 1 of the 2 internal valves is open, if the secondary valve is stuck open I'd expect it to ignite as soon as the ignitor is hot enough.

Questions:
1) How are you determining that the gas valve actually stays open? NG is heavily scented by mercaptan and our noses are very sensitive to the compound. Can you be out at the vent when someone starts it and see if they detect gas very quickly after the motor starts? If it's a short vent it should be within a few seconds of the ignitor beginning to glow.
2) Is the ignitor lit up initially and does it go off? On the 1st attempt when it doesn't light.
3) This was original issue as well?

I should have asked if this dryer had ever been converted to propane or if it has run in this location before? The 2 gases require different orifices and mixing them up causes all kinds of interesting things.

Instead of thermocouples and pilot lights like a gas oven or water heater this has a flame (or radiant) sensor and a hot surface ignitor. This allows it to reset and try again when it fails to ignite, which isn't uncommon. Natural gas (methane) has a very specific combustion ratio needed, 5 -15% by volume of air. If there's too little or too much gas it doesn't ignite. Uncommon would be the valve remaining open to release gas after the sensor realizes there is no flame. I don't recall ever changing a flame sensor, and very few valves.



That isn't the exact diagram for your dryer but the valve is the same. That gas valve is actually 2 internal valves for safety controlled by 3 separate coils. The coil with 3 wires is actually 2 coils inside. If any coil fails the valve it's on doesn't open.

Electricity will take the easiest path, so as long as the flame sensor is closed as drawn the ignitor gets current, as the coils are far higher resistance. When the flame sensor opens the coils get powered. If the sensor was stuck open you'd get constant gas until the thermostat or timer or user removed the neutral. But there'd be no hot ignitor action. Unless it was shorted to ground.

These parts are easily checked by multimeter if you have one; if not they are cheap at any hardware or auto parts store. Resistance values should be close to what is shown in ohms, 1200, 1365, etc... The flame sensor should have no or very little resistance when cold.

If one of the valves was stuck open you'd get close to normal behavior since there are 2 valves. You'd need both stuck open to allow gas to flow through. That would be constant and hard to miss.

I just can't see the combination of faults that could be causing this, particularly since it solves itself after a single attempt each time and doesn't recur until the next cycle.

I'll ask a few co-workers tomorrow when I'm back to work; maybe one of them has an idea I'm overlooking.