It has new tires now though.

There is no cut in acceleration whatsoever, it has tons of acceleration and very powerful 4 low. It only vibrates when coming to a stop, and only when fully warmed up to operating temperature.

The vibration was twice as bad with the Catalytic Convertor, and I considered it undriveable. I am running without one now.

Odometer reads 260k

I have tried:
Remanufactured Bosch fuel injectors
Replaced PCV valve
Motorcraft Spark Plugs
Plug wires with lifetime warranty (no improvement)
Replacing Ignition Coil (one time, no improvement, maybe replace again?)
Upper intake gasket
3rd MAF
3rd IAC
3rd Throttle Position Sensor (and adjusted voltage to .98)
Fuel filter less than 1 yr old
Air filter less than 1 yr old
Replaced fuel pump (starts in 3 turns now so that was good)

Everything I replaced helped it some as a general tune up I suppose, but the problem still remains, vibrates when coming to a stop and vibrates sitting at an idle in Drive, only when warmed up. It does not vibrate when cold. It idles normally in Park or Neutral, cold or hot.

The scan tool has said the same two things 35k before the problem came about:
1) Camshaft Position sensor (Myself, and the auto shop could not align a new one, they said they were having trouble communicating with the computer. I have been running without the Cam Position sensor as there is no Check Engine Light and surely no flashing CEL.)
2) DPFE circuit bad.

Funny thing about the DPFE, I don't know that there is an EGR/DPFE system (am I saying that right?). There are screw in plugs in both the header pipes where the EGR tubes normally go, and an open area where the EGR valve and tubes should be next to the throttle body. I've never seen an EGR valve. State inspection did not fail it though, all they said was to replace the ball joints and tires.

My dad, also a retired engineer, thinks it may be the transmission.

I have a scan tool with live data stream if anyone could tell me how to troubleshoot the Short and Long fuel trims.

 

Take a close look at the condition of the motor and transmission mounts. If they allow the engine and trans to sag in the cradle, there can be metal to metal contact between the frame and engine / trans assembly, primarily around the mount brackets. This can cause unbearable vibrations. Depending on the load at any time, the motor can be cocked or not due to torque and increase or decrease the vibration affect. That could explain why it does it sometime and not others.