Depands what you mean by "unmoving"
Do you mean that you cannot open or close it? or do you mean you cannot move it on the skin fitting?
If the first then go buy a new valve and get a taper plug to fit the skin fitting. Dismantling a valve and cleaning it and getting it to seal are not jobs to be done onboard.
Ayway get a good grip on the skin fitting nut (on the inside of the hull) and with another shifter unscrew the old valve. If it does not move try smacking it bothside (similtanously) with a couple of 4lb club hammers (practice on something else first as it is important you hit the both sides of the valve at the same time or else you are liable to break something. I had to brace the shifter holding the skin fitting nut against the bulk head on me boat and use a spanish windlass to turn the other shifter but it moved and came free quite easily.
unscrew the old valve and push the taper plug into the skin fitting to stop the water.
Now you have time to clean the threads of the skin fitting and wrap them with PTFE tape ready to take the new valve. If not already done grease the new valve before fitting. When fitting the new valve don't overtighten but ensure you can reach teh valve comfortably to open ot close it as required
One final point, no matter where you are make sure there is somewhere near to either beach you boat or have it lifted or prop ot against a drying wall.. As always sh1t happens when you dont want it - little things like broken skin fittings
Wrap an inflatable cushion in a bin liner and position it under the hull, blocking the aperture from the outside. When the seacock is dismantled the water pressure will press the cushion plus liner into the aperture leaving only a trickle coming in.