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I can confirm the 1406 has an electric choke and is 600 cfm; that's what I purchased for my LS4 swap. The carb is configured for the LS family out of the box, so my initial gut instinct is that it is too much carb for that 350 you have, as that is a much more mild engine.
I would definitely recommend getting some tools for tuning your carb. At an absolute minimum, a vacuum gauge. A bit nicer would be a manometer, and really nice would be a wideband O2.
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^Yeah, that's what I was suspecting.
Timebender, I calculate the optimal carb size as 516 cfm based on these assumed specs:
Engine size = 350
Max RPM = 6,000
Volumetric Efficiency = 0.85
That's close to 600 cfm, but assuming everything else checks out, perhaps it needs tuning on the metering jets as the out of the box tuning is for a slightly different setup. If the jets are too big, adjusting the metering screws won't be effective to solve a rich condition.
Given a carb in the right ballpark CFM wise, I think of jets as a large adjustment and metering screws as a fine adjustment.
For tuning kits, see if for Edelbrock 1487 is the right one for your carb.