Steering wheel turning on its own

Your description sounds like the torque sensor inside the steering module has failed and now the steering thinks you are steering to the right at all times. That sensor is inside the lower section of the steering column near the assist motor. Unfortunately it is not available as a replacement part, you need to replace the entire steering column assembly. You can reuse the assist motor as well as the upper switch and SRS assembly though.
An option to help you to drive it until you get it repaired is to unplug the assist motor under the dash. That will stop the motor from trying to turn the wheel. If you remove the trim panel over the drivers foot-well and look up under the dash near the column you will see the EPS control box. There is a 2 wire connector that goes to the motor. That is the one to unplug. These are shown in the first 3 images.

As for replacement, the 03 and 04 Ions are the only ones that use this particular part.
GM 19207619 is the complete column assembly. However you may be able to get just the lower assist assembly from some parts stores as a rebuilt part.

Replacement starts with setting the steering wheel so the front wheels are straight ahead then disconnecting the battery, then you need to remove the trim pad SRS module from the steering wheel then remove the steering wheel itself using a puller.
Next you remove the lower trim panel and the steering column covers (img4). Then you disconnect the SRS and switch module wiring and remove the module (img5). Disconnect the ignition switch harness. Then go to the lower section of the column where the steering shafts connect. Make a line showing where they connect so you can reassemble them in alignment.
Now remove the pinch bolt and slide the lower shaft down off the splines.
Now disconnect the wiring harness connectors from the EPS control module.
Now remove the steering column pivot bolt and the two mounting bolts and remove the column.
Then it will depend on which part you are replacing, if you get just the assist unit you will get instructions with it as to it's replacement. If you get the complete column you need to swap the assist motor and sometimes the wiring on the column. Then reverse the removal process to install it.

Yes, it would become manual steering with increased effort. I would try to get it repaired ASAP simply because both conditions make it more dangerous to drive.