Washing of trucks and trailers is always tough on the road. SEVERAL of the campgrounds I have stayed at had restrictions, (including where we sit today in Williston, FL), Some allowed independent contractors who were registered and approved by the park to bring their "stuff" including water, and do the job on site.
You might want to check around the Jacksonville area for such an opportunity.
Second, along I 95, there are places that specialize in truck trailer wash. Again, along the major highway's only. Once you get inland along the coast, NADA, out of luck. I am sure they will be "pricey"
You could call both Camping World in Jacksonville and Camping World in St Augustine and see what they recommend.
Also final: Pellicer RV campground just opened in St Augustine and is a small park with like 40 sites and the owner is on site. We were booked there in December, But, sudden illness forced us to cancel. I would bet money that for a nominal fee, he would let you wash your rig if you are staying overnight. Personally, I would go that route.