I assume you know it take several hours before cooling starts after the unit is turned on, so it sounds as though you are correct in your assumptions since it doesn't cool with the burner operating, but I would verify it.
I would disconnect the 2 leads to the 120V heating element and connect them directly to a 120V receptacle with an external power cord, bypassing the control board. Leave it on overnight, some say 24 hours. This assumes the heating element is good.
If the heating element is heating correctly and it isn't cold the next day, the cooling unit is bad.
