I would bet that you have a choke problem (not closing when engine is cold) or possibly a bad accelerator pump. You can verify the accelerator pump by removing the air filter housing and looking down the primary (small) barrels of the carb (engine off) while you operate the throttle linkage. If fuel squirts down the barrels when you operate the throttle, the pump is OK. Does the car run OK when warmed up?
Just FYI, it has been my experience that all downdraft Solex carbs are somewhat troublesome, and great benefits in performance and mileage are to be had from switching to Weber or (in your case) a small Holley four barrel carb. A few years ago, Holley was making a small (450cfm) bolt-on replacement for the Quadrajet carbs which came on many GM cars in the 60s and 70s. If your carb is beyond help, you may want to ask Steve B. about this, as I'm sure he'll know how to solve the problem. Good Luck!