Sounds a LOT like the symptoms I was having with my Mustang a few months ago. The lights would dim and the radio would shut off when I applied the brakes while the headlights were on. I had a new Diehard battery, and voltage from the alternator would run up to 14+ at speed.
I finally took it to the auto electric experts for an alternator and wiring upgrade. Turned out that the alternator and wiring were OK, but the alternator (from Autozone), had an oversized pulley on it, and was not pushing any juice at idle. He replaced the pulley and serpentine belt, and all was well.
You can take your alternator back to Advance Auto to have it checked, but if it's like mine it will pass inspection, and still be useless. I recommend a good load test of the entire charging system.
Gregg
2001 Proton Yellow #1379