New TV tires today

What does the door say? Same size just LT?
I recommend ~40 with maybe 50 to 65 in the rear when loaded and towing.
Did they install high pressure snap in or metal valve stems? Regular are good to 65 max.
Check the wheel rating before you go much over 50psi.

35# from the tire shop empty is a good start. If it handles satisfactory, I'd leave it alone and air up for towing.
I run 80# rear and 70# front all the time. Enjoy your new tires.

Proper sidewall deflection is an important safety measure… all tires are designed with a static loaded radius (which is different than the tire radius)… that is the height from the(level) road surface to the center of the axle with its travel load on it… it maximizes the tires footprint on the road…

Weigh the truck and check the tires loading charts is the best and easiest starting point..

No scale, use the SLR measurement until you get a chance to scale it, If that measurement is to high you have too much air for the load… measurement to low you need to add more air to the tire… it puts you in the right ballpark to start with…

Just went through that with my 99 Ram 2500, and it sucked. I had P series tires running 55# all the way around and it handled towing great. Had new Cooper LT "E" series tires put on and aired them to the 55 front 80 rear like the door post says and couldn't keep it on the road. Kept stopping and dropping the air 5# at a time until I got it down to 60# in the rear and it handled a lot better. Still doesn't handle as good as the P series did but that might just be because they are new. Anyway, back to your question, I'd start at where you had your old tires set at and slowly increase the pressure. Of course if you want to spend the time you could always do the chalk check. Draw a line all the way across the tread on the tires and add/remove air until the line is completely gone. More air if it wears on the outside first, less if it wears in the middle first. When it wears evenly all the way across you have the tires inflated correctly for the load.

When you changed out those 'P' tires with MAX of 44# did you check what the rims MAX pressure is?

Upgrading to a higher rated tire doesn't do any good if rims are not rated for increase

My truck is a bit bigger ~7500 lbs ready to roll.

It has e-rated tires and has door post guidance for unloaded and loaded.

unloaded is 40 rear, 50 front
loaded 75 rear, 60 front

I have a little more weight on front being a diesel but might give you some pressures to try.

I do change them when I tow as it really helps to stiffen the back. I got lazy one summer and left them at 75 and prematurely wore out the center of the tread...

Bryan

PS you might google for the "chalk method" of determining best pressure