The Shell petrol station, which you will probably need to visit to top pup your fuel, offers showers for 4 dollars. The best part is that you do not really control who uses them. The showers are in the toilets and are not locked.
So: just walk confidently and straight to the toilets. “Oh, I wasn’t aware of it” is always a great excuse! Otherwise you can only pay for one person and the second or third shower for free.
The Shell petrol station is indicated below on the map and street view.
Not much else is there to say about Fitzroy.
Fitzroy Crossing WA, Australia-18.198105125.567864