You have an AODE.
It holds a whopping 14 quarts of fluid.
You need to use Mercon V fluid in it.
There is a drain plug on the torque converter.
First, have you checked the level of the fluid? Could it be low? You didn't say and that is the cheapest place to start.
When was the last time the fluid was changed? I change my fluid once a year. I am a little anal though.
Here is my story on why I would change the fluid:
I bought my car with about 25,000 miles on it. The first time I drove it, it acted almost exactly as you describe. Shudder, inconsistent shifts, hunting for gears. I bought it anyhow because the price was right, thinking I could always get the tranny fixed.
First thing I did was a complete tune-up and all fluids, including tranny fluid and filter. That completely eliminated the issues. It has not shuddered or hesitated shifting since.
You can get a 5 gallon jug of Valvoline Mercon V from NAPA for like $65 or so, or you can get it by the quart for like $5 a quart. It is unlikely you will need all 14 quarts as it is virtually impossible to completely drain it, but you want to make sure you have enough. I would not recommend having it flushed - in my experience, bad things happen when transmissions are flushed.
When you buy the filter, it will come with a crappy gasket. If the rubber gasket around your pan is still good, just reuse it - it is much better quality.
You might also want to check your TPS voltage. This link will show you how to check it. you need to check it across its entire range of operation to see if you have a "dead spot." Note that if its bad, you need to just replace it. Setting the Mustang TPS Voltage
Finally, there are a few sensors on the transmission itself. If one of those is faulty, you should have a code stored in the computer. Pulling the codes may also be in order to check for that: DIY KOEO/KOER/Cylinder Balance Self-Test Procedure
You are looking for codes in the 600s for transmission sensors / faults.
Finally, just so you are aware, there a some inherent weaknesses in the stock AODE. Accumulator springs, for example, are prone to break. The stock torque converter is prone to expose itself for the piece of crap that it is. And others. That being said, be optimistic, and start with the cheapest things first. May not be the solution, but it is better than buying a new tranny when all you needed was some new fluid...