I tried spraying starter fluid while it was running and it ran, but not for long.
That confirms fuel delivery issue via idle circuit.
Typically when a car sits more than an year your old fuel chips away any rust in fuel lines etc which might be your case . Glad you found filter issue and also carb issue.
However a small crack in accelerator pump diaphragm is not your reason for idling issue. When you give throttle and you see squirt of gas coming out from nozzles is actually your accelerator pump working ok although it may give out in future based on what you described. But still is not the reason for car to die down. Idle circuit is not generally visible to naked eyes, it has path ways inside the carb body that uses engine vaccum to pull gas from float bowls to keep engine running. Even if a tiniest part of rust or debris blocks it , it will create issues idling.
Do this, take out the Idle mixture screw completely out ( I am guessing only one with your Motorcraft carb) and then spray compressed air or carb cleaner with a spray straw it will clear out any debris
Now coming to your vacuum leak given that you are unable to keep it up, how do you know there is a vacuum leak at the base of the carb? post some pics of it. Its quite easy to find gaskets. Is your intake stock given Motorcraft 2barrel?