Welcome to the world of Edelbrock carbs!!
First start after sitting for a few day - common malady. No matter what your friendly manufacturer tech says, that's what they do. If you have the electric choke version, make sure it's clean under the black cover, and is set exactly the way the IB says. If it's a cable operated choke, just make sure the choke system is set right. Next, you really need a phenolic insulator under the carb. When you shut down hot, the heat will transfer through a metal one easier. My personal experience is a 1" thick, 4 hole. It wont cure the problem, but it is a little better.

Stumbling - without knowing what else has been done to your engine - my thoughts are that a .030 over 350 is over carbed with a 750cfm. A 600cfm is really all you need unless you are driving WOT 100% of the time. "Assuming" that all inside that carb is to spec, the step up springs are right, etc,etc, then 'dumping' that throttle to wide open is adding more air, which in turn lowers the engine vacuum to a point where the carb will not draw gas. To operate correctly, you need that vacuum in the carb barrels/ venturi area. Their IB has more info.

You do need the calibration kit to set up your carb - I would probably start at Point 19 on the chart and maybe down one more.

http://www.edelbrock.com/automotive_...404_manual.pdf

These carbs work well out of the box, but IMHO, too big will bite you in the butt every time so you then end up spending lots of time trying to find a sweet spot. And the fuel pressure - you have to have a good fuel pump and maybe a pressure regulator with pressure gauge - and set for 5-6 psig