Tested in Car?
A fully charged battery emits 2.2 vDC in each cell, so fully charged, + to- s hould read 13.2vDC. For charging current to flow into it, the voltage across these terminals must be greater than the battery voltage; normally about 14vDC or so when engine speed is sufficient, which should be always.
Take a voltmeter and check the battery voltage with the key off. Check it again with the A/C off and on, as you describe. Hold the engine RPM at a fast idle and check for improvement. At less than 13.2, your car is taking juice from the cells, and vice versa. I'd want to know if heavy drains, like headlights drop the voltage as the A/C does. If not, perhaps the RPM drop from the drag of the A/C, at idle,
If the alt. and battery checked out, and the voltages above look okay, I'd suspect the cables, but might have both rechecked. The battery should be out, and tested with a 'pile' to load it heavily. Any good electrical shop can test the alternator properly, inclucing the built-in regulator. I don't have a cable test for you.
May we assume that your battery terminals are clean and that the cables are okay? They do go bad, esp. in northern climes.