If your stove works okay, then I do not believe that you have a propane problem. You have a furnace problem. Sounds like you have an intermittent problem where the furnace does not ignite. In my experience that is most often a bad control board in the furnace.
The problem with all of the hot air furnaces we have had is that if the furnace does not ignite, the fan just continues to run until either you turn it off, or if you aren't plugged into power, your battery runs down. From what you have reported that is my guess as to why the furnace is running continuously.
Ed Headington