1997 Ford Thunderbird Cars shuts off, no codes?

First thing I would look at is the fuel filter (because it is the cheapest fix $10), it is directly under the passenger door. You will need the quick fuel release tool to take it off.

Hopefully that will be your only problem, but the smoking after rough start makes me think it is a leaking fuel injector.

Basically what will happen is a fuel injector will stay "stuck" open after the car is turned off and will drain all the extra fuel pressure into that single cylinder.
When you go to start the car after this has happened, the fuel pump priming cycle will not build up enough pressure, and it will crank but not start up. After you crank for several seconds the fuel pump will finally build enough pressure and the car will start, but it will blow smoke (excess gas from the one flooded cylinder).

You can test this easily enough, the next time your car acts like it does not want to start, turn the key to run and wait about 2 seconds. You can actually here the fuel pump prime, turn the key back to off and repeat, you should only have to do this 2 or 3 times, you are just allowing the fuel pump to build pressure.
After you do this, try and start the car. It should catch fairly quickly but will idle rough and smoke off the start, then smooth out once the excess gas is burnt off from the faulty injector.

If the above method works, and you would like to fix this problem, I would first take it to a local quick lube shop and ask them to flush the fuel system, it costs around $100. They will run special solvents through that should clean out the injectors.
If that doesn't help, you might have to look at replacing the injectors, I do not know how much they run - but I assume it would cost at least $200.

Ryan