I have a Maytag Neptune gas (propane) dryer model # MDG9700AWW. When I turn it on, the control panel beeps and the settng indicator lights flash. It was drying clothes but has since stopped doing that as well.

How old is it, and how many clothes do you dry a day? Do you have several people who you dry clothes for? I need a picture of how much it is used. See: https://dl.dropboxusercontent.com/u/8010163/Maytag%20Neptune%20Drying%20Center%20Manual.pdf
I would say that you need to clean it out which means to totally tear it down and clean all lint deposits out. You need to totally tear it down and scrape out and wash all of the parts except the motor, which you vacuum out. Halfway doing it will not work. This includes the exhaust vent, all the way, even the outlet on the side of the house if you have one. You will also have to find the blown safety with an ohm meter. You cannot guess and just change parts. You can go to YouTube to see videos on how to take it apart, but do a very good job or it will fail again quickly. I would get a new belt and change it while you have it apart and get some white lithium grease and grease any bearings you find. You need to clean out a drier every two years, at least, according to the manual. All sensors work together by comparing temperature, so all need to be cleaned. You just unplug the wires from the sensors and remove ductwork that they are attached on and clean the entire thing with soap and water. Make sure you go all through the dryer and get all parts like the blower fan and housing, vacuum out motor well and clean exhaust. Mark all wires you unplug so you know where they go. This is an easier job than you would think. You need a meter to test any fuses that may be blown, which I strongly suspect is the case, but sometimes they will reset if it's an electronic sensor. You will see them on the schematic diagram.
See: http://www.youtube.com/watch?v=A6wmD502-xE and: http://www.youtube.com/watch?v=OBNDbyRm68M&feature=fvsr
http://www.youtube.com/watch?v=a73HTFkcB90&feature=relmfu
Manuals: https://dl.dropboxusercontent.com/u/8010163/Kenmore%20Dryer%20Manuals.zip