LT E tires on a std duty 1/2 ton truck need more than 35 psi as most stickers say.
As mentioned this can and has caused tread squirm leading to sway issues.
Also those big new LT E tires tread is usually thicker when new than a P tire can aslo show tread squirm leading to issues with swaying.
It can take several thousand miles to get the tire broke in and eliminate tread squirm.
I'm not a fan of LT E on std duty 1/2 ton trucks. BTDT and always went back to a good P tire.
I pull a 10k car hauler with 8600-9400 lb on the trailers axles/blue tractor with a cab... with the wifes 2016 1500 chevy 4wd and P265-70-17 tires pumped the rears to max 44 psi. Ran the OEM set of tires for 46k miles and now around 12k on the second set. No sway issues at all.
Play around with psi in those LT E on the rear and see what works better.