Did you get the battery checked yet? The charging system may try to compensate for a dead/shorted out battery.
At an idle the charging is at it's lowest and there might not be enough juice to run the computer, ignition , all electrical and charge battery therfore no run at idle.
At higher rpm the alt is putting out enough juice to run everything and try to charge battery which may explain the high voltage.
2006 unlimited, 2"BB, JKS disconnects, 33" BFG at
1996 Harley Davidson XL1200c