Just to add my two cents. Some stern tubes I have met are not all that thick walled. If large pieces of the drive shaft have detached due to rust of some sort there is a risk that they could jamb between the shaft and the stern tube and tear the walls of the stern tube if the shaft is turned with enough force. The you'd have a very nasty leak to manage.
Since slipping the boat locally is not an option I'd favour towing the boat with the dink and outboard
to open water (after a careful check of the weather
forecast) then sailing somewhere you can slip her after making sure the bilge pump
is good and you have options to plug gaps should the prop shaft fall out.
Less attractive, given the condition of the water, would be to source a short shaft of the correct diameter and push this in from inside the boat displacing the shaft outwards, if the design of the rudder
allows this. Probably need to de mount the engine to do this. I know this is possible on our boat and I have rehearsed it on dry land. Plus I carry the parts
to do it made up from the old prop shaft. I'd hate to do have to do it in the water though, even in the clearest and warmest of conditions. Good luck with whatever you decide to do, you are facing a problem that has kept me awake a few nights.
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