You said that only one breaker trips so unplug the unit and switch the Black (L1) and the Red (L2) on the terminal strip in the unit.

Plug it back in and give it a try.

If the same breaker trips then I would say that the breaker needs replacing.

If the other breaker trips unplug the unit and check the heating element.
Unplug both wires to the heating element.
Check it with a meter, should be around 10 to 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case. This can cause it to draw too much power.

Be sure to put it back the way it was when you are done.

FYI: You have an electrical code problem, at least the code in my area. The two breakers for the dryer should be ganged (tied together) so that if one trips it also trips the other one. Sometimes this is a small rod that ties the two breaker switches together.

In this case not having them ganged may help you trouble shoot but you may want to gang them after the repair.

Here are your parts
Replacement parts for WHIRLPOOL GEW9200LW1 Electric Dryer | AppliancePartsPros.com

Here is a manual
http://appliancedigest.c...ic-dryers-service-manual
You have to join the site to download the manual but it is free and this is a very good site for manuals and other info for the appliance repair DIYer.

And finally see the attachment for the tech sheet

Changing the unit to different breakers seems to kill my bad breaker theory.

Looks like the heater side of the line is the culprit.
I would remove the heater and check the inside of the case for any tell tale signs (weld marks etc.) of the coil sagging and touching the case when hot.
This is a bit of a long shot as usually it would weld itself to the case.

Check for any signs in the heater wiring that it is shorting once the unit warms up.

Check the voltages perhaps there is a bad connection in that side of the line which is dropping the voltage and causing a higher current flow.

If all OK I am at a loss as to what is going on. The only thing you could try is getting/borrowing a clamp on ammeter and monitoring the current flow into the unit.

L1 to L2 should be 240 volts
L1 to Neutral and L2 to Neutral, both should be 120 volts.
If OK
Unplug the unit and check the wires at the terminal strip in the machine to make sure none are loose or burned out
If OK
Check the power at the terminal strip.
Do this with the heater off and on.
[COLOR="Red"]Be careful as 240 volts is lethal !!![/COLOR]