Outdoor shower heads do not turn off. The faucet for the shower head is where you turn them off. But yes, you can get replacement showerheads. Even ones that actually turn off. Most folks I know use this:
RELAQUA AS500W Pressure Boost Spray https://www.amazon.com/dp/B013002H98..._JYakBb01FJYYN
The faucet assembly is about $10 if you need a new one.
I’m going to assume when you say that you are putting water in your black tank you mean that you are using an exterior wall mount flush valve. If that is the case, and you are getting water inside, it is most likely a stuck vaccume break valve. Here’s pics and info on that:
Not trying to be mean but is the switch on? Next suggestion would be the 10A inline fuse. Find it hanging around by the water pump. The pump will run with no water. (not a good idea)
Hope this helps.