Does the icm give crank reference for the starter circuit?

Check if the starter relay is sticking to include the ignition switch

Just to add to this one you will need a test light or multimeter and see if there is power to the small purple wire at the starter when the key is released from the start position and the starter still runs. If there isnt power power at the purple wire with the starter running then bad starter. As far as the code P0340 I need to know what engine you have?

Could be the starter pinion drive is too close to the flexplate-havethe clearance checked- P0340-camshaft sensor

If the purple trigger has power while the starter continues to run with the key not in the run position then that has nothing to do with the pion of the starter. Something is giving the starter power to keep it running.