Jayhawker,
First, we tend to call those little plungers "ticklers", a motorcycle term, really.
Next, that filter needs to go anywhere between the pump and the carburetters. The closer to the carburetters the better.
Also, plain old fuel line is being eaten from the inside out these days by the alcohol in the liquid they sell us with the claim that it is gasoline. As such, it can clog float valves or make them stick open. Older rubber fuel line isn't formulated to withstand this crap. We now use fuel injection line exclusively in place of all old low pressure rubber fuel line.
I'm not completely convinced that any of the stuff so far discussed is what's going on here, though I certainly hope so. Let us know how it's going.
I get the impression that your carburetters haven't been serviced in a long time. Is this so? Try this: Grasp the throttle shaft of one of the carburetters between the thumb and forefinger close to the carb body, but leaving a line of sight to the bushing where the shaft disappears into the body itself. Now apply side force in-out, front-back, up-down, etc. The idea is to see just how much play there is in these bushings. There should be very very little in a good unit. If it seems excessive, look at the other end of the shaft sticking out the other side and watch it as you continue to wiggle.(the shaft!) (Oh geez, I don't know if that's any better!)
Out of shape carburetters are almost impossible to tune, and they cannot hold onto the adjustments you make. Rebuilding and rebushing these babies is not really difficult, but if you haven't done it before, there are pitfalls. Again, let us know.
Motorbill
From Lola to Land Rover, If it's British and has wheels, it's likely I've bloodied me knuckles thereupon