The dryer starts working but after a few minutes the gas shuts off. The flame is mostly blue, with hints of yellow. The igniter glows red hot every few minutes but never ignites again. I have cleaned every piece of lint from the dryer, and vacuumed the exhaust hose. The gas line is not kinked.

All driers will do that when the lint builds up like concrete on the inside of the air ducts; and you cannot always see where it is clogged, but it will cause a safety to open and shut off the heat. You have to totally tear it down and wash the parts out with a good cleaner. That means to take off the heat riser that takes air to the drum and check the filter holder which can be in the door. Then take off the cover to the blower wheel and clean the blower and the casing. Do not miss anything. You will then have to find the safety that is tripping and check it by immersing it in hot water with a thermometer and seeing at what temperature it is opening with an ohm meter.
You need to totally tear it down and scrape out and wash all of the parts except the motor, which you vacuum out unless you want to do some really expensive repairs later. 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 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 all bearings you find. You need to pull the support rollers apart to grease the pins with white lithium grease. You need to clean out a drier every year, 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 (Label wires!) 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/safeties that may be blown. The NTC sensors should be at the intake and at the exhaust air ducts. You will see them on the schematic diagram.
See: Dryer vent cleaning prevents fires - YouTube (http://www.youtube.com/watch?v=A6wmD502-xE) and: Cleaning out a dryer with top lint cleanout - YouTube (http://www.youtube.com/watch?v=OBNDbyRm68M&feature=fvsr)
Cleaning out a dryer: By a woman 2 of 2 - YouTube (http://www.youtube.com/watch?v=a73HTFkcB90&feature=relmfu)

Manuals: https://dl.dropboxusercontent.com/u/8010163/Whirlpool%2027%20inch%20Gas%20%26%20Electric%20Dry ers%20Service%20Manual.pdf
https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20Dryers.zip
https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20G-17%20Gas%20Basics%20-%20Dryer%20and%20Range%20Burners%20Service%20Manua l.pdf
https://dl.dropboxusercontent.com/u/8010163/How%20to%20use%20Test%20Equipment%20Service%20Manu al.pdf
https://dl.dropboxusercontent.com/u/8010163/Electrical%20Theory%20and%20Diagnosis%20of%20Major %20Appliances%20Service%20Manual.pdf