New batteries can be bad too.
Typically one of the 6 cells is bad and loses charge, so it is empty while the 5 others are full and prevent it from being charged.
If you load the battery the bad cell reverses and the voltage drops below 10 volt.
I got mine replaced under warranty and the replacement is hardly better than what came out.
I monitor the voltage (a setting on my UltraGauge) and it generally drops below 12V engine off.
Anyway, I second load testing the battery.
The procedure is simple. With the engine off put on high beams, low beams and the radio.
A fully charged battery could support that for half an hour, at the very least.
A bad battery will fail this test in a few minutes.
(best to run this test where you have a charger or another car and jumper cables just in case)
First to go is the radio. You will also get system warnings. The car won't start if you try it right away (!!!).
Shut off the radio and lights at the first sign of trouble, key off completely and wait a minute.
That allows the 12V battery to recoup enough to make the voltage rise above system warning level.
Key on and start the car. It will. I had this happen several times on my old battery.
Keep it running for a while to allow the battery to regain some strength.
Avoid power off loads from then on.
BTW. During AutoStop the IMA battery keeps feeding the 12V system; no need to fear load on that, but reducing load may prolong AutoStop because it depletes the IMA system less quickly.