I wanted to follow up with my original post. I took the recommendations of everyone here and my fridge is now working
Here's everything I did:
1. Turn off the propane!
2. I cleaned out the flue when I took the fridge out. While I'm sure that was helpful, it didn't solve any problems. (In fact since I still had the burner tube screwed in place it probably would have caused problems if they didn't already exist. So, if you do this be sure to take off the burner tube either before or after cleaning the stack.)
The next two things I did at the same time so I'm not sure which one of them was more beneficial. Although I suspect both of them were at least somewhat needed.
3. I unscrewed the brass burner orifice that is connected at the end of where the LP line screws in (kind of under and to the right of the solenoid) and then soaked it in denatured alcohol for about 45 minutes (probably much longer than needed, but I left it soaking while I did other things). I let the orifice air dry and then gently blew compressed air in it, careful not to touch the small pin sized hole. As people have noted, you don't want to mess with that hole at all.) I then screwed it back into place.
4. I took off the burner tube and "cleaned" it out next. I would have also soaked it in alcohol but didn't have the alcohol with me then so I skipped that step. All I
did for this "cleaning" was tilt it over (after unscrewing it) and watch a lot of very fine black dust fall into my hand. Then I used the compressed air to blow all through it. Then I reinstalled the burner. Note to get the burner out if your fridge is like mine then you will have to first unscrew the large flat outer plate of metal (that has electrical schematics on it - 4 screws), then you unscrew the cover plate that goes over the burner and ignitor (1 screw - have to work to get it off), next you take off one more screw that holds the metal piece in place that the burner tube is connected to.
After reconnecting everything I fired the fridge up and immediately noticed that the flames were all blue coming out of the burner tube. Before they had been more of an orange and yellow color. I also noticed the sound that was mentioned here as sounding kind of like a coffee pot. I stuck a half bottle of water in the freezer and came back three hours later to see ice in the bottle!
So, if you are having problems with your fridge not working on propane then take the above steps (really 3 & 4 first). The whole process, minus the soaking, probably took me 10-15 minutes.
Here is a great site that offers directions similar to what I've written here (but better). http://www.rverscorner.com/articles/refer.html
Maybe if I get time I'll do it all again and take a video this time to post on youtube.