I start the dryer and it runs for about 10 seconds and then makes a vibrating noise for about a minute. This repeats (10 seconds no noise, one minute noise).

Here is the noise (don't blame me for the portrait view, someone else took the video):


I replaced the solenoids and was going to do the valve assembly next, but wanted to check here before throwing more parts at it. I was going to see if I could find an appliance repair shop that may be able to identify it.

I opened up the front and ran it to get a better look. See here:


The flame looks to not be very high.

How old is the unit?
Looks and sounds like the bearing is going out on the drum!

In the second video? The door keeps the drum stable and it was off to shoot the video.

I talked to an appliance part store and they said the first video sounded like the motor was being bound by something. This may prevent the blower from operating and therefore no heat. I would have to find out what is binding it.

If the blower doesn't come on, eventually the thermocouple will cut out the gas.

I took off the vent hose and it is blowing plenty of air, just cold. There is a back vent (I guess for air intake) and I can see the flame not burning very long. I'm guessing there is blockage between the burner and blower but can't find it.

Here is the video with the back vent off. The igniter glows about every 30 seconds. The second time it seems to ignite, but not for very long. It's like it is starved for oxygen?

It's a bad motivator. Get your credits back from the Jawas.

Speaking more from experience with furnaces, but a burner that lights and then immediately cuts off is usually a sign of a bad flame sensor. I think dryers typically use a radiant sensor with a small window instead of a flame sensing rod like in a furnace.