I have a 1996 Lincoln TC. I had almost the same problem, except that at times my car felt like I was driving over rumble strips, when there were none.
I consulted some mechanic friends and did what one transmission friend told me. Go to Ford and buy a bottle of "Fiction Modifier". Cost is about $5.
Then drain all the fluid from the torque converter and transmission. Dont suck it out. Drop the pan and also open the screw in the Torque converter and let it drain. Save the pan gasket, the metal ones are reusable. Replace the filter.
When it is finished draining, put the screw back in the torque converter and close the pan back up. Dont over tighten the bolts. Next add 2 full gallons of fresh fluid, plus the friction modifier. Start the car up and with the wheels off the ground, put it in reverse. You might have to give it a little gas to get the wheels moving. Let the car run in reverse for about 5 minutes. You will see the wheels stop and start a few times. This is normal. The reason for putting it in reverse is that there is only one gear for the fluid to pump through, not 3 if you put it in drive. Worked great for me.
After 5 minutes, put the car in park and check the fluid level. You will probably have to add at least another 2-3 quarts to bring it up to full level. After you do this, again raise it and put it in reverse. The wheels should go on their own and continue without stopping this time. Let it run for 5 minutes again.
Stop the car. Lower and check the fluid level again and top off if necessary. I did this and it solved my problems. Car shifts fine and no more rumbles. My cost was about 2 hours of time, 3 gallons of fluid and a filter. Alot cheaper than a rebuild.
Wednesday, November 14th, 2007 AT 9:51 AM