The fact that one of them is imprisoned in an Imperial-friendly city lends credence to their honesty. Remember, Hammerfell is no longer part of the Empire and has held out against the Thalmor. It stands to reason that they wouldn't be permitted in Imperial-friendly cities for fear of Thalmor retribution. As for being unwelcome in Stormcloak-friendly cities (if such is even the case), I would imagine that has more to do with Stormcloak isolationism than fear of the Thalmor. I've played the quest every possible way aside from slaying Saadia when I first out her, and it seems to me that she's the Thalmor sympathizer, not Kematu
I agree with this.
When Hammerfell continued to resist the Thalmor taking/being given the southern reaches of their Nation, the Empire kicked them out (which was probably one of the most stupid decisions that the Empire made...), which leaves them to be their own Sovereign Nation. Also, Hammerfell is successfully resisting the Thalmor. Alone.
All of this leads to the correct assumption that they are not allowed into Whiterun and other Imperial sympathetic cities. They aren't a part of the Empire anymore, therefore are banned from the Empire's cities.
Because of this, and the way Saadia won't let you know anything even though she desperately needs your help, leads me to believe that the Alik'r Warriors are the ones telling the truth.