I'm no mechanic. However, I've had that happen a couple of times. Are you sure you're not putting the car into the accessory mode instead of turning it off. I know that sounds stupid, but again, I've done it a time or two. Also, it may just be that your battery is shot. I have had batteries go from working perfectly fine to dead from one minute to the next. If it is the original battery, that may be the cause.
This is not a 2005 issue. It is a C-6 issue. I've had my 2012 G.S. paddle shift for less than 4 years. I'm now on my 3rd battery - an Optma yellow top that is now 5 months old. The car sat idle for 8 days with a tender on it and yesterday it was dead as a hammer reading 5.6 volts. A battery disconnect is not the answer either as if you do that you lose everything - seat memory etc. and you have to re-index the windows.
I use Battery Tender too, got a 2 pack on Amazon. One for Vette, other for riding mower. Travel Trailer has one built in. Never again will I go with out one.