If your camper isnt moving around- upgrading can be done when it seems worthwhile. Semi pushing truck w/camper to be expected, a lot of surface area to catch wind. Your bed is probaly a lot heavier gage than mine- so likley OK. T/Ls are considered to be 'cream of the crop' by many but H/J work as well. Given weight T/L cost may be not justifiable.
As to weight, number wise you may be over a tad- but cant imagine to a point of not safe.
Your first post made it sound like camper was moving around. It doesnt sound like anything other than 'first introduction' and nothing, at least from here, reason for concern. Or more like 'relax- your doing ok.
Swapping tires a good move.