"Do you sell any showerheads that completely shut off water flow?"
No, and we are not aware of any showerheads that have the ability to completely stop water flow. With your shower valve still open and the showerhead completely closed, it is possible for hot water to force itself into the cold water line. Once the showerhead is opened again, only hot water flows for a moment, creating a risk of scalding. Showerheads are normally manufactured to dribble a small amount and prevent this from occurring.

"Does the showerhead I ordered swivel, or do I need an additional part for it to do so?"
It is rare for a showerhead not to have a swivel feature unless its position adjustment is in the shower arm or it is a "flat" model that is meant to be installed parallel to the floor. With the exception of those mentioned, some industrial models and most handshowers, standard-style showerheads generally come with a built-in swivel feature.