are non-musty accommodations available in Tahiti/French Polynesian islands?

Greetings,

I'm considering a trip to Tahiti and some of the French Polynesian islands.
I'm allergic to moldy/mildewy/musty environments (and I'm also chemically scent-sitive) ...
and I'm wondering whether I will have difficulty finding/booking relatively economical accommodations that are safe for me in these respects.I would very much appreciate any feedback anyone can offer me about these concerns, as soon as possible.
Many Thanks!

Most economical accommodation will not have a/c so, with the warm humid weather, windows are left wide open and lots of fresh air moving through. I imagine that in the rainy season things could get a little mildewy like in any tropical place but I do not think it should be too much of a problem.