we had the same problem on one of our trailers, the tank's pressure regulator is not functioning properly. ours had water in it at some point and corroded the innards.
And remember as the tank delivers gas there is cooling happening as the liquid turn into a gas, high usage makes it worse, you could be freezing up. (y,es even at 40 degrees). when you use less on the stove it thaws out and your good to go again.
Second thought,,, when your furnace goes out due to low gas pressure it will not re-light until shut off and re-cycled. that's why you wake up cold.