Sounds like your burners are fitted with safety shut offs, like in traditional gas ovens. They rely on a thermocouple in the flame to energise a low voltage magnetic coil to hold the valve open. When I used to muck around with stove conversions you would occasionally have to dismantle a safety to clean the contacts, LPG does contain a small amount of dissolved goo and a well used stove will need an occasional mechanism clean.
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