Re: '56 Continental Hard top
If it were mine I would swell it from the outside with sprinklers along the bottom. Don't like adding that extra weight to the bilge while on a trailer/blocks. Also, the inner planking may be plywood. I'd spray some propylene glycol inside the bilges and let dry.
Is this a varnished hull, TD ?
I believe Sikaflex caulking is reasonably safe to use in the seams as it is easily squeezed out during the swelling process without comprimising the outer planking. Indeed, it will ooze out onto the tarmac if she's out in the hot sun for extended periods. If you are considering caulking please check to make sure of the type to use. There is still the old cotton rope available.
The methods and materials used in the restoration will also play a important role in how to proceed with future maintenance.