Catalytic Converter question??

The Mustang (and many other cars) use a pre-cat O2 sensor and a "after cat" O2 sensor. The pre-cat O2 sensor controls air/fuel mixture. The after cat O2 sensor is to monitor the working of the cats. The rear O2 sensors do not affect driveability.

The Mustang also uses what Ford calls a "light off" cat. This is followed by a regular cat. The purpose of the light-off cat is to allow any unburnt fuel to flare off before getting to the 2nd cat. The light-off cat is not harmed by higher temperatures.

If the light off cat were not present, the 2nd cat could have a short service life due to excessive temperature.

So in answer to your question. The rear O2 sensors are needed to monitor the cats (regardless of how many). If the rear O2 sensors are removed, a MIL will result (unless action is taken).

A suggestion, take it to a muffler shop and ask their opinion and get an estimate. It may not be as bad as you think.

FWIIW, for an automatic transmission car, dying coming to a stop could be a lock up torque converter not releasing as it should. Especially if the car idles fine in neutral.