Try to locate the fuel pump relay and swap with another ' same type ' relay like from power seat accessory, window , powerlocks etc. Better yet buy a new f/pump relay . ANY other relays that are needed to (see wiring diag. )to initiate OR keep pump running are suspect too , in the same way - try a replacement.
Putting a permanent fuel pressure gauge ( install a Tee ) is I N D I S P E N I B L E in seeing pressure at all times. You may add a 4' extension on the Tee for duct taping the gauge to windshield for test-drive observations of f/pressure.
I had a f/pump relay cause all same symptoms , BUT make sure too that grounds at the F/P controller PLUS grounds at any dropping / ballast resister are also sound . Many controllers simply put the dropping resistor "ON" the ground side of the pump - you got power to pump , pump connected to resister which goes to ground . Pump is at a lower ' rate ' . THEN when higher engine speed is achieved - controller TAKES OUT the dropping resister by bypassing or sort of 'shorting it out ' via MosFet in controller. Kind of simple there, kind of bulletproof cause many times the 'dead' controller may be defaulted to - fuel pump 100% utilizing dropping resistor w/no action on cutting out the resistor. That's in nutshell , yours YMMV , maybe a little different ?
Of all the work on cars I do , NOT installing a permanent / live fuel pressure gauge ( my car kinda easy ), not doing this waaay earlier was a self-imposed azz-kicking moment !!!
BUT the elephant in the room is how it ALL happens AFTER rack is worked on