Rainbird Plus
Short answer is yes. When the head is pressurized, it raises. The plus has a slot you insert a screw driver, and turn it off. very effective. You can then manually turn it back on when needed.
I would not just replace the mechanical portion. Dig around your existing 5000 heads, and unscrew the present rotors, replacing with the new "plus" rotors.
It sounds like your assessment for the wettness is the shade. Just a thought, but is this a new problem? If it is, you may have another cause, like a leaky lateral line, a submerged head, or many other easily repairable problems. Is this a low area? If so, and if the wetness is around a lower head, you may have drainage around the head whenever the zone turn off, as in the whole system drains to the lowesr heads. If so, the 5000 and the 5000 plus come with a built in check valve option. Only drawback is without proper winteriztion, freezing may damage these type heads.
Good luck, Jeff