Hi Stang 67,
The old saying is if you think its the carb its probably the timing and if you think its the timing its probably the carb.
Backfiring sounds more like a timing issue and maybe a vacuum issue.
First replace the PCV Valve, it always seems to be the culprit when mine suddenly runs rough.
If you have a timing light, check the setting, but be mindful that if you have an old harmonic balancer, the timing marks might be migrating around - best to disconnect and plug the vacuum advance line (I use a golf tee) and adjust for highest RPM/vacuum.
Then check the vacuum advance line from the carb or manifold to the distributer canister. if its split or cracked its both a vacuum leak and will wack out your timing. get some extra line while youre getting the PCV.
If you are using a stock Dizzy its expecting the spark control signal from the stock carburator, not manifold vacuum, but you said it was running fine, so I doubt thats the issue.
Then it might be another vacuum leak - do you have an automatic transmission or power brakes? Mine had the vacuum lines so cobbled up after 40 years of ham handed mechanics, I built a new vacuum manifold.
I'm afraid if you jump into changing the carb with out checking the vacuum & timing, you'll compound your problems..
mark