1995 Dodge Caravan oil pressure problems

I lost oil pressure. We changed out the pump. Primed the pump via oil sending unit port, and pressure vessel. We have good oil pressure while at a idle. But when the accelerator is pressed we lost all oil pressure. The motor will not build pressure again. We prime the pump and we have oil pressure. We let the motor idle with a mechanical gauge, showing pressure, all the way to operating temp. As soon as the motor speed was increased we lost oil pressure. We put on another pump from a working motor and it behaves the same. Loses oil pressure just off idle. We pulled the pan, and checked the pickup for problems, but didn't see any. Where would the pressure relief valve be, and can it be the problem. The motor has 128K on it. Runs fine, no bad noises like knocks or such. At least while the oil pressure was up. We alway shut the motor off as soon as oil pressure dropped.