Springs and related suspension parts don't normally create speed related vibrations. You either have tire wheel issues, or as mentioned, driveline issues.
Springs cause harsh rides similar to solid suspension, lots of sway, or poor handling, not vibrating. Have the tires balanced again, road force would be best, but any type is better than none. Check motor mounts, exhaust mounts, and rear axle mounts.
On a side note, teflon liners will smooth out a harsh ride, BUT you need good shocks to control the now softer springs. They will still have the same spring rate as before, there is just more slip between each leaf. But, the leafs need to be in good shape, no wear steps, no rust, and all the spring clamps in place. Wear steps is what happens when the lower spring wears into the one above it, creating a step. Over time the springs quit sliding and become one solid mass. When this happens, replacement is the best option. It's not hard to know when you have no spring movement, jump on the bumpers, if the truck doesn't move you spring issues.
Radial tires should be inflated to manufactures recommendations. Side wall markings are for maximum loads, find a vehicle with similar weight as your truck and use it for tire pressures. My '37 has 28 psi rear, 35 psi front, but it has a pretty narrow front tire. Most tire shops over inflate, and most tire issues show up around 40 + mph. I had Yokahama tires on a Toyota Tacoma, they vibrated every time it sat for 24 hours, it took a couple miles for them to loosen up. They did this from the time I purchased them til they wore out, cold weather was even worse.
Joe