liability follows the driver, comprehensve/collision follows car

http://www.claimsjournal.com/news/national/2014/06/05/249762.htm

"Full coverage" usually means comprehensive and collision (in additional to state mandated liability), does not cover your driving another person's car. I think your insurance company is telling you your "comprehensive" and "collision" policy does not cover your accident.

As for why your liability doesn't cover your accident, talk to your insurance again. Usually liability has several hundred dollars deductible, so you might just be better off paying the $840 out of your own pocket to avoid future insurance rate increases.