[wiswind]

Of course, like you, the first thing that I think of is the transmission.
Sticking valves in the transmission......Transmission Range Sensor (TRS) aka Neutral Safety Switch can cause abrupt shifts.
A good idea for a "New to me" car (2nd hand) is to change out fluids.
This would start with a fluid exchange for the transmission.
I have a bunch of pictures posted of various projects on my '96 3.8L......which includes a series that show the fluid exchange.
You can also have this done at many oil change shops.....but I would consider taking in my own fluid if not sure of what they are using.
The correct fluid for ALL years of the Windstar is now a fluid that meets the Mercon V specification....... Look for the MERCON V (not just Mercon) on the label.
Mobil 1 make one.....as well as others.
All you want is a fluid exchange....and the filter changed.
I have used AutoRX (only available online).....but would not use ANY other "cleaner" in the transmission.
The AutoRX (9 ounces) is added to the transmission fluid....and 1000 miles later the transmission fluid is changed.
Transmission fluid needs to be changed AT LEAST every 30K miles on the Windstar.....and is a often overlooked maintenance.......which is not good.
If your unit has many miles.....and the fluid has NOT been changed (or you don't know if it has) and fluid exchange, with AutoRX treatment......and a change again in 1000 to 1500 miles might be an idea.
The thought being......replace dirty fluid.......and the new fluid may work deposits loose......which will need to be flushed from the system.
The AutoRX will help to clean up in the valve bodies.
Of course......nothing will repair wear.......but cleaning things up is a good first step.....

I would also exchange the Coolant, motor oil, and the power steering fluid.
The Power Steering uses Automatic Transmission fluid.......NOT "power steering fluid".

Other things to consider would be normal tune-up items like the fuel filter, spark plugs, spark plug wires and fuel system cleaner (Berryman's B-12 Chemtool).

Getting all the routine maintenance items taken care of may solve the problem....and are things that just should be done anyhow.
I don't want to see you troubleshoot a problem.....replacing things.....only to find out that routine mainenance would have solved the issue.

Oh....don't be scared off by all the projects that I have done......Mine is a '96 that I bought with just under 38K miles on it in '99. Now has over 208K miles on it......and mechanically is in very good condition.