Couple fo quick q's
1. How are you measuring the voltage??? (As in at what point of the system)
2. What voltage has the battery got when the car is running??? (Use a voltmeter directly from the battery terminals)
3. When the car is running is the voltage at the battery going up and down rapidly (by up and down over a volt like its almost ac power)
4. If your getting 14 odd volts when the car is running, then what voltage does the battery drop to when the head lights are switched on?
Im thinking by the sounds of your story so far that even tho the alternator may be producing some power if my have a shagged stater or brushs and hence its not putting out allot of current. So when you load up the system it does not produce enough current and slowlly but surely you run out of power. A fresh battery can last in a car for a week or so with no alternator at all so Im thinking you have no charge at all.
As for the voltage going down that you speak of yes you may have an earth issue, but the above questions will help provide/elimate a possible issue an help achieve a clearer answer.