4 things I would check: 1)Remove Alt and have it "load" checked at your local part store. 14.7V dont mean nothing if it dont have the Amp output to supply the power to the load. 2) Have a "load check" done on the battery 3) Battery Cables, I have seen battery cables corode inside the insulation the full length of the cable and that could be the culprit. 4) Check your ground's, Batt to engine, body to frame ect..
As for the Alt, replace the whole part if it checks out bad that way you know what you got.