The ST tires you have on the TT you recently purchased appear to have given the previous owner good service and may look OK but I would change them out for peace of mind. ST tires tend to not have the same longevity of passenger & truck tires, although they may appear visually OK after 5 years it's safer to just replace them to avoid future mishaps.
I had good service and 4 camping seasons from my stock so-called china bombs and replaced them with Maxxis ST tires which have also given us reliable service going into their 5th camping season. Upon extremely close visual and physical inspection I'm starting to see signs of fatigue although just standing next to them they look pretty good. I'll be looking for a new set of ST tires soon as well.
It will probably be another set of Maxxis ST tires due to my experience with them and the mostly favorable reviews found here and on other websites. I'm hearing good things about the redesigned Carlisle RH tires and I will look at them before deciding. One thing to note is that any reputable tire shop can get any tire you want within a few days, don't just settle for what they have on the shelf.