Since your getting around 12.6 volts and less with the engine running, it indicates the alternator isn't putting out and it probably isn't a bad connection. Depending on what you find when you test further, pulling the alternator is really an easy job. There are only a couple of wires that go to the back of it. I had the same problems you are having. I pulled the alternator and took it to a local shop. They fixed it in less than an hour and it only cost me about $30. It was some kind of resistor and the alternator didn't need to be rebuilt. If possible, I would always take my own rebuilt alternator over another that I might buy over-the-counter.