Chances are it's the brushes in the starter. If it's got over 200,000 miles on it, the brushes usually start to wear out. If it starts up when you wiggle the metal wire (not the battery cable) on the starter, or if you bang lightly on the starter (with a rubber hammer, piece of wood, or your hand...do not use a metal hammer!!!) then it's likely the brushes, since the wiggling/banging moves the contacts ever so slightly, enough for another start.
You can get a replacement brush assembly for 20-30 bucks. Only certain parts stores sell them though. They are sold under the BWD (Borg-Warner) branding. The starter is fairly simple to disassemble, and you can really just pop off the end, remove the old assembly, and pop in the new without taking apart and cleaning the whole starter, though it is a good idea to clean out all the mess from the old brushes.
There is also a chance it's the OTHER solenoid. There are two, one near the battery/fuse box and the other on the starter itself.
You can replace the brushes and both solenoids for a lot less than buying a new starter ($100-200), and the OEM Motorcraft starter is pretty nice compared to the Made in China new ones.