RV tires tend to age out long before they wear out. Just because the tires "look great and fairly new with no signs of dry rot." doesn't mean they shouldn't be replaced. You need to check the DOT date code on the tires to make sure. The conventional wisdom is to replace after 5 - 7 years, Michelin puts the outside limit at 10 years, contingent on condition.
When I bought my 2002 motorhome, the tires looked almost new and the owner told me they were only a few years old. I was more concerned about some other issues so it wasn't until quite a while after I'd brought it home that I checked the DOT code and they were twelve years old and an accident waiting to happen!
Google "RV tire age" for more info. If you do need to buy tires, as long as you stick with a name brand, you'll be OK. You don't need to buy the most expensive tire on the market. Most of the tire recommendations on this and other forums are based on personal opinion or "my brother-in-law uses XXX tires at his trucking company". It's kind of a Ford vs. Chevy thing. After reading some of the tire postings you'll have a pretty good idea of what brands to look at.
Also, try to find a well-regarded tire shop in your area that serves the trucking trade. I found one that also serves many local RV dealers and followed the tire shop's advice, saving quite a bit of money in the process.
And make sure the tires the shop is installing have a relatively current DOT code as well. This is more critical with RVs than cars since car tires wear out sooner. Personally, I'd prefer 6 mo or less but up to a year should be OK (ask for a discount). If you have a spare, instead of keeping your best old tire, you're better off buying a relatively new, used tire for the spare.