Sounds like the accelerator pump cup seal in the carburetor has hardened or cracked and is not giving enough pump shot to the venturies when accelerating! You will probably need to overhaul the carb. What engine does this mustang have 351C?
Start with the basics. Your car requires a clean air cleaner...cleaner than fresh white toilet paper.
If your air cleaner looks kinda like it has been used once or twice (that brownish tinge) buy a new one - a couple of bucks at Wal~Mart.
Spark plugs only last 12,000 miles(a year) before they start not working perfectly. Replace them.
Ignition system (points, cap, rotor) also wear out. Spark wires, if they look good and new should be OK(unless they are the originals) as they age too) Get a complete set.
The carb "might" be sucking air through the vacuum lines. Buy a spray can of disc brake cleaner and while the engine is idling , spritz the rubber hoses that connect to the carb or manifold and see if the engine idle speed changes in any way. Stalling or speeding up is an indication that there is a leak there(where it is suppose to be tight). Fix by cutting the hose shorter and sticking it back on as the original end that was connected is cooked rock hard(so not grabbing to the outlet pipe on the carb or manifold. If you need new hose - take the old one in the auto-parts store with you and buy new hose. These are simple fixes anybody can do and my fix it up right there and then.