When the lines were ran during construction the holes that were drilled through the wood were too tight. As the pipe expands and contracts you will get a ticking noise that sounds exactly like a drip. As the seasons change and the humidity of the house changes the sound can come and go.

This is your answer.

Where you adjust the temperature is the cartridge that is being recommended that you change. The valve that changes the different heads is just a diverter valve. You don’t show any views of the temperature valve but I caution you disassembling without trying to identifying the manufacturer and having the replacement cartridge on site. Nothing worse than removing a cartridge and having it come apart and not being able to reinstall and having the water off. Looking closer I can tell that the brand is Kohler. You should be able to look on their website and identify your exact model and pick up a cartridge locally.