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Once worked on a Ford Explorer with a similar problem. When it was cold out at about 35, it would fire, but quit. If I held the gas down it would run, and as the engine got warmer, I could let up on the gas. When I put my computer on the Ford, found out that water temperature sensor was stuck at 72, so computer was trying to run engine to lean, and as engine warmed up it ran fine. Had to replace the water temperature sensor.
First off is this a diesel of Gas engine? If you used either and it did start, but only ran for a few seconds, you probably have a fuel problem. Have you changed the fuel filter? Could also be a weak fuel pump.
Most Ford explores have a vacuum operated valve in the heater line to the heater. It is located on the passenger side towered the back of the engine in one of the heater lines. You should be able to move valve with your hand. Once engine is up to temperature, feel both sides of heater hose before and after valve. it should be hot on both sides if water is flowing. If not move valve and see if it gets hot on both sides. If that it is the problem, just prop valve open until you can replace it.
Sometimes if fuel pump pressure is to low, a car will not start. Try taking off one of the vacuum lines to the intake manifold, and give it a shot of earthier, then put line back. If it starts or runs for a few seconds, it could be a fuel pump or even a clogged fuel filer problem.
My first guess would be a bad ground or positive connection, or wire. If problem happened when you replaced the battery your connections may not be as clean as you think. I had someone use Vaseline on the battery terminals to prevent corrosion, and just that thin film of Vaseline caused the same problem that you have.
My Guess would be that something is wrong with the rack and pinion unit. I have seen this happen on some Dodge mini vans. Turn one way easy and not the other.
Could this be your problem? I worked on a Explorer a while ago and when it was cold it would crank, but not run. When it was warm it was fine. Problem was that water temperature sensor was stuck at about 72 deg. So when vehicle was cold like 30 deg.,computer was trying to run engine too lean and it would not run.
What is the code? Most of these GM cars have a computer chip in the key, and if the wires inside the ignition break it will not start. To confirm this, connect a jumper wire to the starter solenoid "small wire" terminal. Touch positive connection on battery and if engine will crank, then problem will be confirmed. To repair wire you will need to take ignition out, and splice small orange wire that is broken. No need to replace ignition assembly.
Could possibly be a vacuum leak. The plastic intakes on these engines have been know to leak.
What engine do you have? If it is the 3.8 V-6, the oil pump is behind the front harmonic balancer, and it is inside the timing chain cover. It can be changed without taking the engine out. There are special instructions on priming the pump.