Angus
You have the old Groco rubber plug type seacock. They were a top notch unit.
The Tee handle must be loosened a tad to rotate the plug. A bit of water will come in - normal - when the Tee is loosened and the plug turned. Once retightened the weep stops.
Unfortunately they are no longer available. I've looked from time to time when in a used marine goods store and have never seen any that I can use.
The taper type seacocks that started the thread, in addition to being adjusted properly, the need for grease now, they may also need to be lapped lightly are also top notch units.
Use of a light Clover compound, lapping, will lightly grind the two tapers and make for a good fit.
The compound must be thoroughly washed off with solvent and rags. Any grit remaining will continue to grind eventually damaging the taper. Not hard, just need some attention to details.
And yes, they should be exercised periodically and greaed to be sure they don;t freeze.