As a general rule of thumb, I believe you can use the max pressure and the max load for each tire to compute the pressure you need.
Take the Goodrear 245/75/16 OEM tires on my '98. The max tire pressure is 80 PSI and the max load is around 3000#, it means that at 80 PSI, each tire can carry 3000# of load. My 2500 could carry 6000# of weight on both rear tires, or 12000# weight total, if I could load up the front end. (This exercise completely ignore axle and spring capacities. )
Unladen, my truck weighs about 6000#. Dodge recommends 55 PSI front, 40 PSI rear, unladen. That makes the weight distribution 58%/42% front/rear, or 3500#/2500#; this works out to 1700#/1250# front/rear per tire. If I then put 3000# of horse dung in the bed, this changes the distribution to about (guessing) 3700#/5050# front/rear.
A simplified (reduced) formula might be:
unladen pressure + (max - unladen pressure)
* (current tire load - unladen tire load) / (max tire load - unladen tire load)
So, if 40 PSI is good for 1250#, and 80 PSI is good for 3000#, then the proper inflation for the rear tires whilst carrying a 5050# axle load should be:
40 + (80-40)/(3000-1250) = (x-40)/(2525-1350), or about 67 PSI.
Now, my 19. 5" Hankooks are good for 3640# at 95 PSI (3415# dually). With my 6. 5" wheel width, the best tire pressure seems to be about 75/65 PSI front/rear, unladen. The same load of horse dung would require rear tire pressures of:
(95-65)/(3640-1250) = (x-65)/(2525-1250), or about 81 PSI.
This calculation should put you in the ballpark for correct tire pressure whether loaded or unloaded.
An alternate method: if your current tire pressure gives you good wear and performance and proper tread contact, measure the 'radial bulge' at the bottom of the tire. Then put a load on the truck and inflate the tires until the 'radial bulge' is the same. You'll then be pert near the correct tire pressure for the load you are carrying. Alternately, you could measure center-of-axle to ground, then inflate until you reach that height again. Come to think of it, this latter measurement method might just be more accurate, because proper tread contact happens at a certain loaded tire radius, regardless of load. If you can't reach the desired loaded radius without exceeding the tire's max pressure, you're probably carrying too much weight. In this case, your rear leaf springs looking like upside-down 'U's would be a strong hint as well.

(With 4000# of horse dung in the bed once, I wasn't yet near the suspension stops, but the springs *were* somewhat inverted

; and I was still well within the wheel, tire and axle limits. )
N