389 Starving for Fuel??

Hi there. I am a new Pontiac fan (great timing huh?) and new to the forum also. Hoping someone can help with my issue.

I recently purchased a 1959 Catalina, all original, with a 389 2bbl engine. The car had been sitting for ten years. After pulling the fuel tank, flushing it and the lines, and rebuilding the carburator, the car fired right up. It seemed to be running great in the driveway. I put a new set of points in the car and set the timing. Car was running even better now. I was stoked over how great the car was running and decided to take it around the block. Just as the engine built some RPM's, just before shifting into high gear, ol' Jersey (that's the cars name)fell flat on her face. Acted just like she had ran out of fuel. I could keep her running with a tap of my foot but just barely. It took a while for her to come back. After a couple more stop and go's I decided Jersey had a bad habit. This issue was happening every time at about the same exact spot! Adjusted the float as high as I could and it did not seem to change the problem any. Jersey has two fuel pumps, a mechanical and an electric pump. I don't know much about this set up, so any information on this would be helpful. I have only found a couple articles on the fuel system and from what I can tell the double pump setup is stock. The pressure of the mechanical pump read about 6 lbs and the electrical read about the same. Pressure seems okay I think. Any advice??? Oh yeah, one other thing, she does the same thing in park if I bring her RPM's up and hold them there. After a few seconds (5-10 approx)the engine falls flat, like running out of gas.