Your Google directions are pretty good, there isn't really any way around Jax that doesn't involve a lot of traffic, unless you exit 295 and take US17 south on the west side of the St John's river. Take 17 down to SR16 at Green Cove Springs, then take 16 all the way over to St Augustine.
If you don't want to get off 295 onto 95 Southbound, stay on 295 for 1 more exit and get off at US1 -its a nice road and nicer scenery (IMHO) than 95. Take US1 (also called Philips, then Old Dixie Hwy) all the way down to St Augustine. Take San Marco Ave to May St/A1A (an old carousel at the corner?) and go left/east towards the Vilano Causeway/bridge. Go over the bridge then left again at the coast on A1A up to the North Beach Campground. Vilano beach is quite built up (from when we started camping near there 40+ yrs ago) and there is a Publix grocery store and several restaurants at the corner.
As an alternate scenic route exit off US1 just south of 295 at the Nocatee Pkwy and head east to A1A then south to the campground.
We leave in 4 weeks for St Augustine Beach but since we are approaching from the north (Ontario) we take 95 to 295 on the east side of Jax down to Hwy 9B then cut over to US.1 then over to A1A. By the time we get to Jax we have had enough of I-95!