After the car warms up, after about 15 minutes I apply the brakes and as the car is slowing down and is almost at a stop, there is a thud that you feel under the front of the vehicle.
Noises can pinpoint all over a car. Go on Youtube and watch Scotty's video. Do a search for Finding the Sources of Car Noises Scotty. He'll show you how to use a device to help pinpoint the exact location of the noise.
then on the fourth time it will do it again plus a burning smell in the front left. If I really have to apply the brakes
Sounds like something in your brakes is worn out. Have you checked your brake pads/rotors/etc for wear?
he did not provide me with the code numbers but stated they meant the Shift solenoid was at fault and the pressure control was at fault.
If you take go to any parts store they'll scan it for free. Unless you give us the precise OBD codes it's very difficult to point you in the right direction.
I have taken it to four shops, and they have no idea. Two were Ford dealers who claim they never felt the thud.
Who/what were those 'shops' you went to? Were they chain shops? Also dealers and chain shops often employ part-changers rather than those skilled at diagnosing problems. It is in your best interest to find a good independent mechanic in your area (guide in the FAQ) who will diagnose rather than load up and fire the parts cannon.
sounds like a binding driveshaft