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On a street engine, I'd say you definetely want vacuum advance hooked up for better throttle response.
On a drag car where you are launching at a predetermined RPM, you already are at or above your mechanical advance point. It is not as important to advance the timing right off idle.
Since this problem just started, I'd look for stuck float/ needle and seat or power valve like 12 sec Stang recommended. Since your fuel pump is pretty new, I wouldnt worry about it yet. But I wouldn't rule it out. New parts can fail too.
When is it running rich? All the time? Is the other end of the PCV system open?
You're getting way too much info. Use the KISS method. Disconnect and plug the vacuum line going to the distributor. Set the engine idle to 800 rpm. Set the initial timing to 8-10, ?? I forget. Then connect the vacuum advance up again. It should be connected above throttle plate, no vacuum on idle. Burn way too much fuel at part throttle without vacuum advance
You're right on the carb, have to set the floats off the car but unless you took it apart and played with them they should be ok. You can set the idle with a vacuum gauge. Turn the idle screw in til it starts to stumble and back off a shade. Give it a rev to clear up and then slowly back off the screw until the vacuum no longer increases. Check engine rpm and adjust as required and then do the other screw. Recheck rpm and adjust if required. Then check the timing again and you're done.
Who knows what distributor you are using, what cams or springs you have installed in it etc. I have a bag full of different Ford limiters and weights. Setting it for max advance at RPM may not be good thing when everything is unknown. My 55 Pontiac is like that but I'm running a fixed dist. with a timing box
He mentioned back fire but didnt specify thru tailpipe or intake. Regardless the first issue is rich, my first asumption would be blown power valve, if your confident all else is adjusted right. There are some models that require you to turn the mixture screws in the opposite direction than normal. If your timed vacuum port is not flowing at slightly off idle throttle possition than its plugged and i would suggest fixing it and hooking the dist there no where else.