No the starter doesn't need replacement. If the starter was bad it wouldn't even crank.
If the battery is new and was at low charge I would say your alternator is on it's way out. It could have barely passed the autozone test. Have the retest it. I would say that is your problem if it isn't you could have a bad wire in between the battery and the alternator.
I recommend taking it to the Ford stealership or a local shop. Describe your problem and instruct them to inform you of their diagnoses prior to doing any work. Post what they said is wrong and we can give you further advice on what to do based on your abilities, cost, etc. | 
