Hi Mick - sorry to be late with this but not being a DVC member, I rarely visit this board.
The A1A runs all the way to Miami, dipping inland from the coast at Jupiter Island and then you'd have to join Hwy 1 for a short distance until you can turn left onto the A1A at West Palm Beach.
I think you are going to find this quite a slow lengthy trip and would suggest taking the A1A to Fort Pierce and going on the Turnpike to Fort Lauderdale from there.