Your dryer is doin what I call misfireing ... The part shown below is the cause of that ... Replace the coils and Im sure your dryer will fire every time your ignitor cycles.
Coil kit for dryer gas valve (2 coils, new style) - Made by Whirlpool. This popular, newer style solenoid coil kit fits all dryers with two-coil gas valve. Brands include Whirlpool, Kenmore, Maytag, Admiral, Crosley, Frigidaire, Estate, Amana, Kitchen Aid, GE, Hotpoint, Roper, White Westinghouse, Sears and others.

Part Number: AP3094251
made by WHIRLPOOL Our Price: $14.87
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The heat is regulated with a Klixon mounted next to the blower housing near the vent pipe so it can judge the temperature of the air coming out of the drum.Right next to it is a melting type fuse. If the dryer runs and gets too hot the fuse will melt.But the dryer will still run. So if the fuse is good(continuity) and the dryer control(cycle) thermostat is also good then you should have power going to the gas valve assembly when the machine is running unless the timer is in the no heat or wrinkle free/cool down mode. When the gas valve assembly is energized the flame sensor

mounted outside the burner chamber detects the heat from the ignitor. When the heat is very intense then the sensor shuts down the ignitor



and that frees up the power so that the coils can be fully energised and open the gas gates. The gas comes out and hits the red hot igniter and you have ignition. It stays on for maybe 2 or 3 minutes or until the control thermostat is satisfied and shuts down power to the gas valve assembly. So...
If your gas dryer has a good flame sensor and a good igniter AND the dryer will work some and quit, replace these:


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Replacing them is easy. All you need is a phillips head screwdriver. Make sure the quick connects are tight. And do not touch
the ignitor when doing this operation:

The coils get hot and quit on these machines when they get a little age on them. It is aggrevating because it is a hard to diagnose thing. But well worth the 20 bucks.