You set the thermostat where it has always been. The thermometer reads correctly. Still the house feels damp, sticky, and heavy. That is a different complaint from too much heat, and it is worth separating before you touch anything.
What the feeling is telling you
A cooling system removes heat, and as it does, it pulls moisture out of the air across the coil. When the system runs long enough, the air dries. When it cycles off quickly, moisture stays behind. A room that holds its target but feels clammy is often a room where the air has not been dried enough. That points at run time and airflow, not at the setpoint.
How to tell it is humidity
Place a simple hygrometer in the main living area and log the reading for a few days at different times of day. Note the temperature beside each reading. If the moisture reading runs high while the temperature reads right, the complaint is humidity. Note whether the reading climbs after cooking, showers, or a full day of people indoors. Those are the loads you are working against.
What to look at on the system
Check the air filter first. A dirty filter slows airflow across the coil, and the coil then does less drying. Confirm that supply and return registers are open and not blocked by furniture. Check the fan setting on the thermostat. A fan set to run continuously moves air without the coil doing its drying work, so the house can feel damper. Note whether the system runs in short bursts or in long, steady stretches.
What to ask before you change anything
Ask whether the system was set up to run long enough to dry the air, and whether anyone has checked the airflow since it was installed. Ask whether a dehumidifying option exists for the equipment you already have, and what it would require. Do not replace equipment on the strength of a feeling alone. Log the readings first and let the record guide the decision.
The central air system's typical service life and planning notes are on the Central Air Conditioner page. Use it to decide whether the complaint is a comfort problem or an equipment one.