Most Cougars from 1983-88 came equipped with delay windshield wipers, as it was a very popular option. Most of the time they work great. But, with time and usage, they can start acting very strangely: sometimes the delay function doesn't work, sometimes the wipers won't go all the way down, sometimes the whole thing won't work at all.
If your car is a daily driver (not a show-quality vehicle, in other words), then a few well-placed whacks on the wiper motor with a hammer could get the ol' wiper motor jump-started. Now if the motor is bad, it's because one or more of the internal contact brushes on the motor is disintegrated. Banging the casing with a hammer will only temporarily solve the problem. You'll probably need to either buy a new motor assembly, or find a used one at a salvage yard or whatnot.
But with a little disassembly, you can find out a few things and possibly fix the problem very easily. The "park" contact in the connector housing can get gummed up. To fix it, remove the screws holding the halves together (NOT the wiper motor itself). Drill out the two rivets holding the housing on, pull it away from the base, and remove the pin holding the rocker for the contacts. Under the rocker there are two removable copper contacts and springs; remove them and clean all contacts with fine sandpaper or a Scotch-brite pad. Then, reassemble everything in reverse order (you MUST pop rivet the housing back onto the base, however). Then put the two halves back together and you're done. Should keep you going for a good long time.
Along with the physical wiper motor there is a wiper regulator module (aka governor) located under the dash, on a steering column support brace, to the left of the column. It controls the electric functions of the wipers including delay and shutting down. A replacement module was around $85 new the last time we checked. Usually a salvage yard piece will function fine but be warned that replacing this module is not fun. In fact, it's easier if you drop the steering column to get to it.