Hi Jason,
While your computer and polargraph are still on speaking terms, hit cntrl-c to open the controller's console. Issue commands using the controller until it the polargraph gets stuck on 'busy'. Was there an error message of interest in the console?

Motors/voltages: (Warning! - I'm not an expert, and there are subtleties I'm ignoring. This is a general guide only) If that 4.1 ohm spec is correct (you can check the coils with a multimeter), and you're giving them 5V, using Ohm's Law, each motor should be taking about 1.2A. That would put you over the 2A of your supply. Is your supply getting hot? Are the drivers getting too hot to touch? If everything is merely comfortably warm, you should be OK.

If you've got the power jumper removed from the motorshield, I think your uno is safe.

I'm doubtful the 'busy' freeze has anything to do with power. It's usually because the polargraph has received a command it doesn't like -- it'll return a message like "coordinate out of range" (something along those lines) in the console. That's not unusual when first setting up a machine. It's easy to get one of the machine specs wrong, or do something with the controller that's unexpected, resulting in commands that stymie the polargraph. And that usually happens when the pen is moving because the polargraph will be told to move the pen somewhere it thinks it can't.

The fact that your motors and drivers are cool is unusual, though. You sure you've pulled the power jumper out of the motorshield? I could see there being a problem if you've got the power coming from the uno and instead of your adapter, causing the uno to reset or something.

Anyway, do the console check before going out and buying a new adapter. (And check that the jumper is off the shield).

Okay, I reproduced with motors running.

incoming: READY
regenerating preview queue.
regenerating preview queue.
Dispatching command: C09,3550,3541,END
Last command:C09,3550,3541,END:2031137544
regenerating preview queue.
incoming: SYNC,3550,3541,END
incoming: READY
incoming: READY
incoming: READY
incoming: READY
incoming: READY
incoming: READY
regenerating preview queue.
Dispatching command: C01,4345,4445,END
Last command:C01,4345,4445,END:1514302686
regenerating preview queue.
incoming: SYNC,4345,4445,END
incoming: READY
regenerating preview queue.
Dispatching command: C01,3856,3944,END
Last command:C01,3856,3944,END:711519010
regenerating preview queue.
incoming: SYNC,3856,3944,END
incoming: READY
regenerating preview queue.
Dispatching command: C01,4933,3155,END
Last command:C01,4933,3155,END:133239904
regenerating preview queue.
incoming: SYNC,4933,3155,END
incoming: READY
regenerating preview queue.
Dispatching command: C01,5415,3916,END
Last command:C01,5415,3916,END:4267995948
regenerating preview queue.
incoming: SYNC,5415,3916,END
incoming: READY
regenerating preview queue.
Dispatching command: C01,4271,5171,END
Last command:C01,4271,5171,END:3160939933
java.io.IOException: Device not configured in writeArray
	at gnu.io.RXTXPort.writeArray(Native Method)
	at gnu.io.RXTXPort$SerialOutputStream.write(RXTXPort.java:1171)
	at processing.serial.Serial.write(Unknown Source)
	at processing.serial.Serial.write(Unknown Source)
	at polargraphcontroller.dispatchCommandQueue(polargraphcontroller.java:2652)
	at polargraphcontroller.draw(polargraphcontroller.java:698)
	at processing.core.PApplet.handleDraw(Unknown Source)
	at processing.core.PApplet.run(Unknown Source)
	at java.lang.Thread.run(Thread.java:695)
regenerating preview queue.

It stays busy with the last command. Any ideas?

I am running OS X 10.9.3, so I had to install the legacy ftdi drivers. They've never given me any problems, but I mention it in case there's a known issue or something. Sounds like the controller and server are losing serial connection.

I will try a different USB cable.

I did post this problem to the adafruit forums and they had a few suggestions to try, including pulling one h-bridge chip out. I was able to replicate the error with only one motor running (since I only had one chip connected). When I pulled out both chips (empty sockets), but leaving the motor shield connected and powered, I could not replicate the error.

This points to a problem with the shield in my mind. The issue is that I don't see why the motor shield should cause problems with the arduino's serial connection if they are separately powered. I verified that the light goes out on the shield if I unplug its power, so I'm very confident I'm not trying to power steppers off of USB.

Think I figured it out with Adafruit support Bill's help. The heat sinks atop my stacked chips were shorting the pins. When I pulled the heat sinks off, I could not replicate the usb port exception.

Bottom line, the heat sinks slide around a little when you're pressing on them to get the chips seated in the socket. You need to make sure they are not touching any of the pins on either side.