Customer welcome to just Answer!.
Code 37 is one of those codes I've learned to disregard in these years because the circuit isn't used, like a code 34 would be on a vehicle not equipped with speed control.
What you've described very well is what I'd expect on a vehicle that's losing fuel pressure, big blue. As fuel pressure and volume decrease, the fuel system's ability to deliver enough fuel to support higher engine loading is the first to go... you'll find the engine is a bit "soft", losing power and surging. Eventually it won't pull itself in any way, but may still idle. And eventually even that will pass.
Since the same power source feeds the injectors, ignition coil and fuel pump, it doesn't seem likely you have an intermittent loss of power to the pump... at least from the source (the ASD relay). The relatively slow nature of the stall suggests a pump that's very weak and can only support the 15 psi of fuel system pressure it's designed for for a short time before conking out.
If at all possible, buy or rent a fuel pressure gauge that can be placed in-line at the throttle body with your fuel supply hose. The hose size is 5/16" and you'll need a couple feet of extra EFI hose to do the testing, along with the needed clamps. Drive it until it starts acting up and check your pressure. If it's falling, momentarily squeeze the smaller return hose coming off the throttle body in the same area. If it makes no difference, the problem is that the fuel pump is stalling out, not that the regulator is stuck wide open.
If you need to replace the pump... it it seems pretty likely... let me know if you need any help.