Cables and control heads require maintenance and lubrication, preferably done BEFORE they show need of it.
Most things mechanical that I install on my boat get disassembled and liberally greased first.
Cables like a product called “Tri Flow”, it’s not too thick or sticky, and seems to have some silicone in it.
I make a funnel out of plastic and duct tape around the upper end of the cable housing, flood it, and let the lube soak in while I rock the cable back and forth for a while.
Comodaves recommendation to check the components individually is sound, control heads can be limbered up, and cables are easily replaced.
A sticky cable merits replacement, you can waste a lot of time trying to make an old one new!