You should lie down in the shade and rethink this. That's vintage cast aluminum, they don't make 'em like that anymore. It's not broke, you are the only person even notices it. That can't be the ugliest thing on your boat demanding attention is it?
Good advice once he sees the replacement cost for a new shift.
There should be a set screw on the bottom of the handle. Put the handle all the way forward or back and the screw should be visible. If the screw comes out, then the handle should just slide off the shaft, although the corrosion may make this a bit challenging. Once off, try polishing the handle. If that doesn't work, then the handle
@Tally Ho or a similar one should work. If the handle doesn't come off, then your options get more limited, live with it or by a new shifter.
The button in the center of the handle should push in so that the engine speed can be increased without engaging the transmission. If that button doesn't depress or is very hard to depress, remove the whole unit and try cleaning it up and inspect the cables while it is out.