We recently bought a set of Target's Casaluna brand of linen sheets and absolutely love them. They are a little rougher than our cotton sheets, but the coolness is far superior to any thing I've ever used. They are just a bit over your price range, but meet all your other criteria. I would 100% recommend these, we are considering a second set.
https://www.target.com/p/100-washed-...05#lnk=sametab
And I avoid any sheets that aren't cotton or linen. I don't care what kind of coolness claim they make, nothing is as good as natural fibers.