Yes, you’re right about regular carpenters glue. It’s great stuff for always-dry applications, but it turns to mush when it stays wet. That’s why you really only have two practical options:
- type II or type III weatherproof PVA wood glue
- polyurethane glue (PL Premium , Gorilla Glue and Excel are popular brand names)
I’d use either of these with confidence, though the PVA is nicer to work with because it’s water-washable before curing. Also, I’d supplement each overlapping joint with a single wood screw. Stainless steel is definitely the best, though hot dipped galvanized is okay, too.