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The gas smell can come from 2 different things. Either your gas valve is bad as you suspect, or if you recently had some work done on your house like floors being refinished or new paint or anything like that, the dryer will collect and condense the fumes and it'll make your clothing smell like ethanol. There isn't much you can do to fix the second issue other than wait for a few weeks for the fumes to go down. Otherwise if that isn't the issue then the valve will need to be replaced.
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Gas coils can't cause a gas leak, you'd need the entire valve.