You have a bad fuel pump.
"new" no longer means "it works"
"new" now means "never tested"
Simple test would be to manually turn on the fuel pump and then go for a test drive, this would eliminate the computer or wiring shutting off the fuel pump during acceleration.
In the engine compartment fuse box is the Fuel Pump Relay
Turn key OFF
Pull Relay out, you will see 5 slots
If you have a volt meter or test light, test the slots, only one of the slots will have 12volts, that is the Fuel Pump Power from the fuse, it is active all the time.
That is slot 30 on all automotive relays
If you look here:
http://www.reuk.co.uk/Automotive-Relays.htm
There are diagrams of relay sockets
You want to jump slot 30 and slot 87 together, you will then hear the fuel pump come on.
Go for a test drive, if your engine still stumbles then you will need to replace the pump, new or not.
Fuel pressure and flow are related but not the same.
Say your engine requires 1 pint of fuel per minute at full acceleration/load.
And only 1/4 pint per minute at idle
Say your fuel pump is bad and can only deliver 3/4 of a pint per minute
At idle pressure will be high and engine will idle and Rev fine
But when under load as demand gets above 3/4 pint per minute fuel pressure drops off fast and engine is starved for fuel.