How can I fix my Firebird 90 oil burner?

My 20+ year old external oil burner is misbehaving: It starts up only if absolutely cold (first thing in the morning) but shuts down after an hour or so and will only start up again if left for several hours. Reset button does not seem to help nor does turning up (or down) thermostat. Seems to me like it must be a faulty thermostat which is misreading temp and shutting down and looks easy and cheap to replace (sits on top of burner with two wires and temp dial). Any advice?