If you replaced the flasher on your 1999 Mercury Cougar and your turn signals are still not working, you should check the turn signal fuse. This fuse is located under the dashboard on the driver's side of the vehicle to the left of the steering column.
If this is the case, try disconnecting the ac compressor. If not that, it could be in the 'master switch' (turn signal lever unit). Does anything else go out along with the turn signals? Does it only happen when you turn on the heater or ac?