Having no interior heat on the 1997 to 1998 Lincoln Mark VIII is a common problem. The problem is a part that Ford calls A/C Electric Blend Door Actuator ( F7LZ-19E616-AD) that costs only about $25, but requires over 6 hours of labor to replace. The cost to install this part at a dealer is $600 to $900. It would be far better to make the part so it does not break so easily. It is advised to reinforce the part so that you may not need to replace this part so often. See Blend Door Actuator Reinforcement for a procedure to reinforce this part. The following procedure to remove the old actuator and install the new one takes 5 to 6 hours. With practice, you may be able to do it in 3 hours.
Remove the four screws that hold down the A/C Electric Blend Door Actuator. Three of the screws are visible. The fourth one is in the upper left hand corner. A 1/4 inch ratchet wrench is useful for removing these screws. I also found that a 5/16 inch GearWrench worked great for getting at the hard to get screws to remove and then later install them in the new Blend Door Actuator.