Well unfortuantely, given the inherent nature of leaf springs, that's just not gonnna happen. It's impossible to have a spring/shock combo that will carry as much weight as you expect it to without sagging, and still ride that cushy when unladen. The only way to achieve that is with air, and that's expensive and complex.
I have yet to ride in a pickup, brand new or otherwise that does not have the rear end chatter at least a little over bumps while completely empty (I haven't ridden in enough new, coil-sprung Rams to know how they act). It's nearly impossible to have the shock valved stiff enough to handle the load you ask it to, while still giving you the unloaded performance you want. Quality shocks will greatly improve things, but unless you're willing to spend a lot of time and money, you'll have to accept a sacrifice on one end or the other. You have to remember, you're asking a pickup to ride like a car while empty, or while carrying an additional 1000#. That's a pretty large gap you expect it to excel at.
You could probably run closer to 30psi in your tires and that would help significantly. A stiffer carcass tire like what you're on will ride harsher while unladen than a P-tire will, but it will also support itself better and probably allow you to run a slightly lower tire pressure. Look up how to chalk your tires properly so you can find the ideal pressure to wear evenly and ride nicely. Don't go lower than 30psi or it'll get squirmy and you will begin to sacrifice handling.
You may be able to get satisfactory performance out of it if you're willing to spend on a set of aftermarket leafs and rebuildable shocks, but you're looking at $2000 in parts for the rear alone to go that route, probably more.
Anything can happen for enough money.
