There is nothing wrong with the distributor and the ignition system works great when in good condition and set correct. What happens is the points or plugs or plug wires wear out and people start playing with the carburetors to fix the ruff running motor, but the point gap is off or something else is actually off making the problem which when that is off that changes the ignition timing, or the plug wires are old and shot creating resistance in the wirers and the firing of the spark plug. Weak spark is a big problem with tuning these cars. Get metal wires and make good clean connections. The gig is to get all the ignition stuff done first then set the valves cold at. 010 and then see how the car runs with choke in and warmed up.
The choke system on a tr3 can be a real bummer and let the fuel jets stay open and flood in more gas and make your symptoms also. Look at the jet and make sure it has fully returned after the choke is in and the motor is warm say ½ hour of running. These chokes kinda work backwards of other old cars in that they let more gas in and not shut off the air to choke the motor.
The ball would be an aftermarket needle float assembly that people put in for higher performance with the fuel. You would need to lift the lids off the carburetor float bowls and look to see what you have.
It is always best to do a full tune up to know where you are at: plugs, points, condenser, and new distributor cap, plug wires the whole system then do the carbs last.
steve