Classic/Antique Car Repair: 1963 Fairlane Gas Tank, 1963 ford fairlane, good luck jack

Question
I have a 1963 Ford Fairlane. There are a couple of issues im having with my fuel system.

1. The gas tank is full but the car thinks that it is empty. The car will "run" out of gas but when i go to fill it up it will only take 1-2 gallons. Sending unit of the fuel pump bad?

2. Now my filler neck is leaking now. Can i just replace the filler neck or do I have to replace the whole thank?

Answer
I work on Model A's, but I will try to help out.  
Here is what I think:
Your tank is full and your pick-up is missing or misplaced,  using only the top 2 gallons of fuel.  
Your filler tube grommet is being tested with the full tank and it has dried out and leaking.  Somehow, get all the gas out of your tank and pull out the sending unit and take a peak.  Remove the filler neck, not sure how, and check out the grommet.
good luck.
jack