...then start the car (with all accessories off). If the voltage remains at 12.6V or just slightly over, then you have an alternator problem. A good charging system puts out 14.4V and nominally, 13.5V.
If the alternator is the culprit, your model year contains the early alternator unit that has the removable voltage regulator/armature brushes component.
This is a good thing as you may not have to replace the whole alternator. When the brushes wear out, your alternator can't do it's job effectively...it's an $85 fix versus $250...