Are there public showers to use after running in the park if you're just there for the day?

No you have to be a campground resident.. Yea "Broom" runs that area.. Beware of Broom he don't like trespassers... Better off jumping in the lake.

I have not seen any when I went there!

Definitely there are showers, but as said, I think they are only for campers. (How did you like the event last weekend?)