That 7psi is high, but depending on the carburetor's seat tension might not be causing any trouble. But if you have a carb with smaller floats that are not putting as much upward pressure on the seat, perhaps that's just enough.
But would you not be able to see more fuel than normal dripping into the throttle bores while it's running?
Alternately, if you just turn on the pump, do you see it filling up the bowl and leaking out the vents, or forcing itself into the throttle bores?
If it's not doing that when the pump is running constantly without the engine running, then it's likely at least not the main problem.
But since you can't drive it, you really don't know how well it's running. And in fact it sounds like it's running like crap from your description!
Popping and cracking and backfiring is anything BUT normal or good.
Sooty plugs too.
And adjusting the idle air mixture screws does little to nothing for the overall mixture. You may be running too rich. Time to check what size jets you have. If the carb is brand new and out of the box, then you should be able to look it up. If it's ever been messed with, you should verify visually.
Agree with the comments that, while the carb is perhaps larger than you need, the size disparity alone does not cause these issues.
Did you ever check the quality of the spark at the plugs? If you lay a plug wire (prefferrably connected to a plug) out on the engine is there a massive and loudly snapping white/blue spark? If not, your ignition is weak.
And what did you set the timing to this time? Do you know for a fact that your timing marks line up with the engine's internals? You might just try advancing the timing more anyway, but knowing where your timing marks are in regard to the piston is critical on any engine, and these old Fords are unfortunately not blessed with consistent external markings.
Paul