It's the second floor. Moisture shouldn't be an issue. I would forgo the moisture membrane, glue it, and staple it where the flooring will be parallel to joists. I would be afraid without glue the subfloor and flooring would flex at different rates between the joists and create squeaks.
I'll agree with others in recommending replacement. I would actually recommend fiberglass instead of steel though. It won't dent or rust like steel will. Also, pay the extra for composite jambs. The bottom of the jambs always rot and composite will prevent that.
At the risk of sounding like a smart***, doesn't it make more sense to order $30 worth of wood than to drive 3 hours round trip to buy it for a slightly better price or even receive it as a gift?
Welcome to the forum! I don't know if they are looking to hire anyone, but I have done some business with these folks and they seem like good people:
https://hmwoodworksnc.com/
Welcome, Linc! I met you a while back at Bills house when he was kind enough to help me resaw some hickory I had. Glad to see you found the forum, though I'll have to pretend I didn't see the New Englander at heart part 🤣