How did Gandalf sneak into Dul Guldur?

[The Squatter of Amon Rûdh]

As for why Thr*áin was not relieved of his map and key after capture, Gandalf surmised that Sauron only wanted the last of the Seven that Thráin kept, and once he had it he merely 'flung the broken prisoner into the pits to rave until he died'. It makes sense to me.

This is the source of my problem: Sauron only wanted the ring, so his search would be fairly cursory; but surely no-one would suggest that Orcs are averse to robbing prisoners of their belongings, or of stripping them naked to make sure they've found everything. Just look at what they did to Frodo when they captured him and their subsequent fight over the booty. Imagine what they'd be like if Sauron had already taken what he wanted.
As I said the disguise working makes perfect sense, given what we know about Orcs from the trip to Mordor in RoTK. A pair of dark glasses would fool that lot.