I do it various ways, depending on the gun, bird and cover. In thick cover prefer to hunt with round (or rounds) chambered, and safety on. Prefer a non-automatic safety on doubles, and most can be easily converted by removing the "auto-link." Want to shoot again after reloading, if that's possible (and it often is with some upland birds, early in the season) , without dinking with the safety again.
My 686E doesn't have an auto-safety and I prefer it that way as well. Nothing else I use puts itself on safe, so I would rather it be my habit than the gun's.
But have also hunted quite a bit in more open country with outside-hammer doubles. With some I could NOT cock both hammers with my thumb, so was limited to cocking one hammer--and then trying to cock the other quickly IF a second shot was possible. With those I usually hunted with the gun open, with both hammers cocked, then snapped it shut while mounting the gun. This worked well on more open-country birds, but not so much in thick cover.
But my most recent hammer double, a J.P. Sauer 2-1/2" 12-gauge made in 1911, is not only delightfully light and lively, but I can cock both hammers while raising the gun. This seems pretty safe to carry--as did the Model 97 Winchester pump I've hunted with the chamber "hot" for many years, leaving the hammer on half-cock.
But outside-hammer shotguns aren't very popular anymore....
I think I'd be fine on solo walkabout with charged chambers, on safe. If hunting in a group I might choose times to leave the action open and walk with charged chambers. I admit to not trusting the tang safeties (mind you I don't 'trust' any safety to keep myself and others safe). They are right where you might handle the gun and inadvertently wipe them off.
I don't have any exposed hammer doubles, but have an Encore that I think would be fine with a loaded chamber. The hammer doesn't rest on firing pin, and I doubt it would be easy to cock by accident.
