I've stayed in Little Manly Cove, it's just around the corner from North Head. Balgowlah is also good, it's nice and sheltered from all but easterlies.
There are no public moorings, but any free mooring can be used, so long as you don't leave the boat in case the owner returns.
The various yacht clubs will allow you to stay overnight on a berth for a few dollars, so it may pay you to give them a call to see if it's worth it to you, for the ability to have a hot shower before bed, or before you leave.
Have a safe and happy trip!