You almost surely have an air leak in the vacuum side of the system. This would often not be large enough to allow noticeable amounts of gas to leak out, but enough to kill the vacuum sufficiently that the pump doesn't pump very well. This also permits any gas that gets out of the tank to run back in once the engine is off. The leak could occur at any connection, or in the flexible hose connector. The first step would be to make sure all connections are tight-teflon tape is a good safety move on these. Then carefully inspect the flexible hose-if it is original modern gas may be eating through, or it could just be tired. The most insidious leak source is where the tube goes into the tank to pick up gas-if this is above the gas level, it can act just like an air leak anywhere else, and there will never be any evidence of it.
As you have just replaced the pump, the connections to it would be the prime suspects. It could also have an internal leak.
I replaced the original AC pump on my '53 with a Carter in antiquity. After 120K + miles I have just rebuilt it. I never had any problems whatever, including the dreaded vapor lock. Air leaks are likely the primary cause of fuel pump issues-if they are not adequate to stop you by themselves, they will team up with other factors such as high underhood temps to do it.
When inspecting your lines, look for either particularly clean areas, indicating gas leaking out, or dirty areas, suggesting wet areas picking up crud.
