Here is how the whole thing works. In winter the furnace warms the air and the blower blows air in your home while in summer the AC evaporator coil removes humidity and heat. The condensator line carries the surplus of moisture to the floor drain. At this point, the compressor circulates refrigerant through the evaporator coil and expels heat outdoors.
Furthermore, the refrigerant travels through the evaporator coil and the compressor evacuates heat outdoors. Your home is then supplied with cool or warm air. Finally, air is sucked inside your house and carried to the furnace where it is conditioned. Humidity is added in water and the air in the house is cleaned before being recirculated.
You should always replace both the furnace and AC together
You might be tempted to install only the system that is giving you a hard time and wait to purchase the other one. Your air conditioning unit works together with your heating system to provide optimal temperature. You will only be lowering the performance of both systems. You will save more money when you change both as your furnace and AC share the same air handler.