Wouldn't they say Shams instead of Damascus?
Nope. Shams means "sun"
Maybe you mean Shám which is actually a name for the whole Levant region in Arabic but Syrians use it for Damascus and its surroundings. So it is better to stick to the formal meanings usually.
I was expecting Ash Shams from having seen this in other courses. But Damascus makes more sense.
Ash-Shams الشمس = the sun.
Damascus is sometimes referred to (specially by Syrians) as Ash-Shám الشام.
This name particularly is used sometimes to note the Levant in general.