Sounds more like the ECM temp sensor isn't working correctly. Make sure the connector is plugged in and the wires are good.
If you've got an autozone that will scan your vehicle for free, drive it over there as cold as possible, and check the ECM engine temp. Once the engine warms up it takes it a LONG time to get back to ambient temp, if the sender is not working correctly only down low, it may only noticeably "act up" when the engine has been sitting overnight. If you don't have a scanner, kind of hard to get a baseline cold engine check.
Or if you can find the resistance values chart that exists for the temp sender, use that to determine if the sender is accurate at lower temps.
Pretty unlikely it would run correctly under WOT with an inadequate pump, if it even put out enough pressure to idle the motor. Which most (if not all) TBI pumps do not.