Decks, Patios, Porches and Docks - covered patio addition

I've recently had my concrete patio extended and now want to cover it with basic osb and rolled roofing. It spans 16' with 14' to the post x 30'. I have 3 post footings with planned 1'x1' columns which will get foam wrap and stucco. It will connect to a wood frame 2 story home. my questions are a) how big of a header should be used to support the 15' spans? and b) would 2x6 run the 14' to header or should I upsize to 2x8 or 2x10? I'm going to run cross beams at 4' intervals to help secure osb. I live in phoenix AZ if that changes anything. Thx for all help in advance.

Spanning that length, a minimum of 2x10 will be required. You may opt for LVL for strength. Another concern I would have is just what pitch will this cover have? That span will have to be 12' or more high at the house in order to maintain a decent pitch for water runoff and allow a good height at the trailing end. Placing the LVL on 24" center and crossing it every 4' will provide support for the OSB.