Will, do you have any pictures?
I can pull out my sending unit WITHOUT doing anything to my fuel tank.
No need to pull it...
You said 61 falcon... I have TWO 61 falcon's here. Maybe you have an incorrect fuel tank.
That's the filler neck and the tank vent.
The sending unit is in the center if the tank directly behind the 3rd member.
Pull off the 5/16 fuel line there and let the tank drain a few gallons out into a clean bucket...
If you can get it all then you don't need to take the tank out.
If you cannot, then put a cap on the tube to keep gas from draining everywhere when you lift it.
Then pull the tank, UP and OUT of the trunk.
Set the tank on it's side, then remove the sending unit and replace your float.
If your fuel tank is bolted on the UNDER side of the trunk then I have no clue what tank you're working with...
FE