The exhaust on my '67 122 wagon smells strongly of gasoline. When the car is parked and off, there is no gas smell. Also, the car cold starts perfectly, but when I have been driving it and shut it off for a few minutes, it has trouble starting and sometimes puffs white smoke for a second or two, then runs fine. It isn't very fuel efficient right now either.

Is this an issue with the carburator? What is the best approach to diagnosing the problem?

White smoke is not gasoline. It might be oil or coolant. But the symptom
you describe indicates a rich mixture or overfilling of the float chamber.
You should probably examine your spark plugs and see what color they are.
This will give you some clues as to what the problem really is.
Might also pull your float lids right after shutdown and see if they
both have the same amount of gas in them. Should be about half full or maybe
a little lower. Also check to make sure that your float valves are tightly
screwed into the float chamber lid.
--
George Downs, The "original" Walrus3, Bartlesville, Oklahoma