In answer to your question- Gas will flow to the carbs as long as your tank petcock is open. It is a simple gravity feed system.
The two most likely causes of your problem are -
1. Float levels set too high. ( you have checked this) or
2. Float needle valves are not seating.
I have a 72 350F that I rebuilt this winter and my carbs worked fine last fall when they were removed. This spring when they were re-installed I had the same issue as you are having. I did the same checks as you and even examined the float valves ,and thier seats, under a magnifying glass and they looked great. Just because I had them out I tried cleaning them with a scotch brite pad (fine grade equivilent to #00 steel wool)
I cleaned both the needle points, and the seats, and it seemed to do the trick. There must have had a thin coat of gasoline varnish on them even though they looked good. Hopefully this helps you.