It sounds to me like you are having a solenoid problem. The solenoids ensure that your transmission switches from gear to gear cleanly and holds that gear until the conditions change and powertrain either downshifts or upshifts.
At one time, an automatic transmission relied solely on mechanical means to shift — bands/clutches, gears, torque converter and the rest — and that was it. As more precise shifts were required both for the best mileage as well as top emissions, the automakers began using electronics to take over the jobs that the mechanical devices handled.
For example, instead of relying solely on the clutches and bands and gears to make sure the right combinations were selected, today’s transmission uses a series of electronic sensors — the throttle position sensor and mass airflow sensor, as well as others — plus a number of solenoids and the valve body to ensure that the right gearing is chosen.
Let’s assume, for example, that your Mustang is shifting from 1st to second. At one time, it would take a combination of the right rpm point from the torque converter plus the clutches picking the right gear timing the upshift through the use of the clutches bands and valve body, to make sure the upshift happened. For years, it worked reasonably well, but it wasn’t precise.
As more precision was needed, the automakers added sensors that watched, among other things, the position of the throttle, the pressure flow through the manifold, as well as the air flow at the intake. There were also sensors at the wheels and on the crankshaft watching to make sure everything was in sync for the proper upshift.
Successful upshifts are enabled by a number of solenoids or switches that divert the transmission fluid through the proper galleries in the valve body. All of this must work properly to make sure your car upshifts and downshifts correctly.
Now, you have lost third gear and the way it is acting, it is likely the solenoid is either sticking or has failed completely. I would run a diagnostic looking for the right codes that indicate a solenoid failure and I believe you will find the OBD test will show you which solenoid has failed.
You will find the solenoid packs are located externally on the valve body or internally. If it is an internal solenoid, you will have to pull the valve body, which means a partial transmission teardown and replace the proper solenoid pack. If it is external, which it may well be, then you are looking at simply swapping the proper pack yourself.
If you can repair it yourself, you are looking at about a $200 investment for the part. If you have to have a technician tear things down far enough to get to the internal solenoid, you are looking at about a $1,200 bill.
I hope this helps. Please let me know what happens as I would like to share your outcome with our readers.