The valve is better known to just fall apart when it fails, whereas these are classic heater core symptoms, especially if you haven't used it in a while or kept up on coolant changes
The slider isn't mechanically connected to the valve on the auto climate control. The cable goes to a servo. Better off to check the valve position when going from hot to cold or manually moving the valve.
But it sounds like your heater core is restricted and needs to be back flushed. Does the...