Same problem...three times before but never again
I had the same problem and did the same things: fuel injection clear, replace FP, replace TFI, etc., you name it I did it.

The Stang would go through the same sequence: hard/long start, rough idle, popping under acceleration and ultimately stalling even after I did the changes.
What I found to work (in my case) was ensuring the computer had a good, solid ground. Most sensors are pigtailed in one way to the body. Sometimes the body gives offers too much resistance and does not provide a sufficient ground connection or either signal return back to the computer.
Check/clean all of the grounding points on the block and the body. Ensure the flat braided straps that connect the block to the body are still in good shape and not frayed. If all else fails, ground the signal return wire on the self test connector to the negative pigtail right behind/near the battery (not directly to the battery).
I'm not saying this is the only cause

, but after fumbling with this problem for nearly a year and almost wiping out on the highway, this worked for me and hasn't happened again since. Besides, if you're running the 3G alternator, having a clean, low resistance ground will improve your voltage readings/output.