About this:
in their human forms, they don't know if another person is one of them or a "normal" human
I don't know if they know for certain - but Baran (Mith) recognised a 'vaguely familiar' scent about the assassins who were in Gondor.
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Cami
My camel source says that camels walk at @ 2.5 mph; trot/jog (very easy for them to maintain over long distances) @10mph; and can do short distances at a canter @ 17mph.
So 6 or 7 hours east of the Umbar city is around 70 miles.
From there to where the Eagles are along the foothills of the mountains south of the mannish city is about 250 miles
So how about 3 days of travel from the oasis - giving time for pit stops and sleep.
How's that sound?