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Justin Case Auto
1st step is to see what the coolant temp. sensor is reading when it is cold. Not the gauge on the dash but the sensor for the computer. If it is reading too high the vehicle will not get enough injector on time for a cold start. Basically the computer thinks the vehicle is already warmed up. If that's ok next is to test fuel pressure. If it's low again the vehicle will not get enough fuel flow for a cold start.
If the pressures are good when it goes warm and the outlet pipe from the evaporator is still cold at that time it may be a problem with the ATC system in the vehicle.
Have a shop smoke test the intake system for vacuum leaks
I agree with the two above but because the battery is drained when starting cold have it tested also, it may be starting to get weak.
In my opinion... we go with factory convertors when ever possible but I have had luck with the Walker brand. I agree with Pushrod, you may want to test/change the O2 sensors first. We alway but in new O2's with new convertors so sometimes we will change them first with the understanding the customer may need a convertor also. We do all the O2's upstream and down stream.
First is to check the key fobs. The metal tab that holds the battery in will come loose a lot on these GM fobs. If it is get the fob repaired or replaced first.
If the intake manifold has not already been updated to the new one do yourself a favor and replace it with the job. You'll be better off in the long run.
Check for dirt in the tank. If it looks clean I would go to a different brand pump. You should also check the electrical connector at the pump, make sure it clean, tight, and not burnt.