In response to some of the questions/comments:
1)The hot water heater and stove work fine. (We don't run the fridge on gas)
2)I will turn the thermostat lower at night. I'm not man enough, though, to wake up to 40 degrees :)
3)There is no noticeable smell of gas either outside or inside the unit. I have checked (and will recheck) all the exposed connections, though.
4)I'm not sure if the furnace cycled or ran continuously during the night. It was cycling before we went to bed, but after that I was asleep.
5)I will check to see if there's soot by the exhaust.
6)The tanks were all filled by weight and charged per pound.
Thank you everyone for the information, replies, and advice. Maybe I was just expecting too much out of my furnace. My camping season isn't going to begin this year until May (so using the furnace may be a non-issue), but I'm still going to try the different suggestions. If I do figure something out, I will update.
I appreciate everyone taking the time to help me out. I hope everyone has a fun and safe camping season.
Thank you all again.