Here's a more systematic approach to troubleshooting:
1. Check the A/C System Pressure: Low refrigerant (R-134a) is the most common reason for an A/C clutch not engaging. Even if you *think* your refrigerant level is fine, it's crucial to have the system's pressure checked by a professional with the proper gauges. Low pressure is a safety mechanism to prevent compressor damage.
2. Pressure Switch: A low-pressure switch senses refrigerant pressure. If the pressure is too low, this switch will prevent the compressor clutch from engaging. This is a common point of failure.
3. Cycling Switch (Compressor): Some vehicles use a cycling switch on the compressor itself. This switch prevents the compressor from overheating by cycling it on and off. A malfunctioning cycling switch could prevent the clutch from engaging.
4. Clutch Relay (Part of a larger relay assembly): While there isn't a dedicated A/C clutch relay, the A/C compressor is likely controlled by a relay within a larger power distribution center or under-hood fuse box. This relay might be part of a larger module controlling multiple functions. You need to consult your owner's manual or a wiring diagram to locate and test the relevant relay. *Simply visually inspecting a relay often won't tell you if it's bad.* You need a multimeter to test its continuity.
5. A/C Clutch itself: The clutch coil within the A/C compressor could be faulty. This is less common than the pressure switch or low refrigerant, but possible.
6. Compressor itself: The compressor could be seized or failing internally.
7. Fuse (Indirect): While there isn't a dedicated fuse, a blown fuse in a related circuit (like the power to the A/C control module) could indirectly prevent the clutch from engaging. Check all fuses related to the A/C system in your fuse box and under-hood fuse box.
Recommendation: Trying to diagnose this yourself without a wiring diagram and a multimeter is difficult and risky. Take your Lincoln LS to a qualified automotive technician or A/C specialist. They have the tools and expertise to properly diagnose the problem and avoid potentially damaging your compressor. Simply replacing parts at random is inefficient and expensive.