OK, Matt, now we're getting somewhere.
First, Lincoln did a very bass-ackwards thing with the Mark VIII; while all of Ford's other model lines switched over to OBDII Networked Restraint Systems (some as early as 1995), the Mark VIII kept a stand-alone, "count-the-flashes" RCM right up to the end of its production.
That's why you got no codes with the OBDII scan tool.
-Code 51, "Thermal Limiter Blown"
This is not a conventional "short blows fuse" scenario.
The RCM has detected a shorted wire that could trigger one or both of the airbags to deploy at the wrong time. If this happens while the driver is just cruising down the road, minding his own business, Whoa! Accident! Injury! Death!
To prevent this, the RCM takes a cyanide pill and kills itself by overloading an internal circuit and burning out its deployment power fuse.
The RCM (which is in the middle of the dash, behind the radio) must now be replaced, but only after the underlying problem is solved.
Where's the short?
Your RCM is telling you with the alternate code you see. You've reversed the digits; it's not making a 31, it's making a 13.
-Code 13, "Deployment Circuit Short to Ground"
I'll copy'n'paste Ford's explanation.
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"DTC 13—Air Bag Circuit Shorted to Ground
NOTE: DTC 13 will flash only while the short to ground is present.
The air bag diagnostic monitor measures the voltage at Pin C276-11, Circuit 614 (GY/O) and Pin C276-12, Circuit 623 (P/W) of the air bag diagnostic monitor connector. The voltage at these pins varies with charging system voltage. If the air bag diagnostic monitor measures a voltage of 2.0 volts or less at Pins C276-11, Circuit 614 (GY/O) and C276-12, Circuit 623 (P/W), the air bag diagnostic monitor will flash DTC 13 to indicate a possible short to ground on these circuits; refer to Possible Causes for additional circuits that may be shorted to ground. When flashing DTC 13, the air bag diagnostic monitor opens its internal thermal fuse. This disables the air bag deployment circuit. If the voltage at Pins C276-11 and C276-12 returns to normal, a DTC 51 will be present due to the open thermal fuse; refer to DTC 51 in this section.
Possible Causes:
-Low voltage at air bag diagnostic monitor Pins C276-11 and C276-12.
-An internal short to ground within the air bag sliding contact, causing the driver side air bag circuit(s) to be shorted to ground.
-An internal short to ground within the LH kick panel safing rear air bag sensor and bracket, causing Circuit 614 (GY/O) or 623 (P/W) to be shorted to ground.
-An internal short to case ground within the driver side or passenger side air bag assembly."
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Roadboss has already pointed out the single most common reason for 13/51: the vehicle has been in an accident, the airbags went off, and the RCM was not replaced.
The Code 51 Diagnostic in Ford's manuals begins like this:
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"DTC 51 After Air Bag Deployment:
NOTE: Air bag diagnostic monitors can withstand several air bag deployments and do not need to be replaced after every deployment. Only replace the air bag diagnostic monitor if it is damaged.
Occasionally, after an air bag deploys, the internal wiring of the air bag(s) may become shorted to the metal housings of the air bag(s). This internal air bag short is detected by the air bag diagnostic monitor as short to ground in the air bag deployment wiring. Since the air bag diagnostic monitor is still operating immediately after most deployments, the air bag diagnostic monitor will detect the shorted wiring and will flash DTC 13 or 14 and open the internal thermal fuse. After the deployment, as the air bag(s) cool off, the internal shorted wiring may correct itself. The short to ground will no longer exist and the air bag diagnostic monitor will flash DTC 51. If a vehicle with a deployed air bag is flashing DTC 51, inspect and replace all the damaged areas of the vehicle that have crushed wiring, sensors, etc. If no damage is found, assume that the deployed air bag was the cause for the intermittent short and replace the air bag diagnostic monitor when the new air bag module(s) are installed.
After a short to ground has been corrected, the air bag diagnostic monitor will flash out DTC 51. DTC 51 indicates that the thermal fuse inside the air bag diagnostic monitor is open and the short to ground no longer exists.
If a short to ground has not been located and serviced (short to ground corrected itself), consult OASIS for potential wiring shorts.
If a short to ground has been correctly located and serviced, replace the air bag diagnostic monitor.
*** DO NOT replace the air bag diagnostic monitor until the short to ground has been positively located and serviced. Replacing the air bag diagnostic monitor before servicing a short to ground will result in repeat service.***"
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Other than that, look for an internal short within the clockspring in the steering column (I see that fairly often) or corrosion/water damage at the safing sensor behind the driver's kickpanel down low at carpet level.
If you like, follow the wires referred to in the Code 13 diagnostic. They only go to both airbags and the safing sensor.
If you're using an ohmmeter to check resistance to ground, NEVER NEVER NEVER NEVER probe the actual wires going into the airbags!!! Always unplug them when measuring deployment circuits. The power sources in most DVOMs are enough to detonate the 'bags!
Let us know what you find.