A bad battery shouldn't cause low voltage while the car runs. That sounds more like your voltage regulator, else a bad connection. Also, check the terminals at your alternator and battery.

Recently, I had charging problems on my 78 firebird, shortly after installing a 'one-wire' 100amp chromed alternator. Thought the new alternator had prematurely kicked the bucket, but it turned out that the heavy wire to the alternator had broken most of its conductors at the ring terminal. The few strands which remained were not sufficient to supply the current required to run the engine, lights and stereo.