There are several user problems described in this post..
First a description of how the Stay-A-While and the older Extend-A-Stay work.
They have a series of check valves in them (one for each port) to prevent gas flow from one tank into the other or the escape of gas if one tank is not connected.. But which tank supplies the gas is a matter of which tank has the greater internal pressure.. Normally this can be either tank depending on ye old Mercury (Temperature) the hotter the tank the higher the pressure.
At night the hotter tank will be on-board, and the furnace is burning gas.
This is why you shut off the main tank if you want to run on the AUX only.
Second we have the question of "Why did the burners not go out"
Well you have that hose,,, The hose from the aux tank to the Stay-a-while is full of gas at high pressure,, Not sure how high but over 100 PSI. The line pressure in the RV is 11" of water, or about 1/2 PSI as I recall (I have done the math but do not feel like re-doing it today) it is going to take a fairly long time to burn off that 100+ PSI hose fill, that's the problem.