well it seems like you're going after another issue. If your car was still "dumping fuel when it's off", it would have issues starting the 1st time as well, not only the 2nd time. While your injectors might benefit from being re-wired, I can almost guarantee that it isn't the issue you described originally. They aren't letting fuel through when the car is off, otherwise it would never start for you, ever.
Based on your original post you've got a sensor issue, not an injector issue. The first two that come to mind are MAF and Coolant Temp Sensor, neither of which you've checked. With those two being left untested, I'm not really able to help since you really aren't being thorough in your troubleshooting. Testing sensors is easy, rewiring injectors is not as easy and has the potential to give you more problems stacked over what you've got going on currently.
Also, it will be much better if you test and compare to the FSM for your vehicle/engine, not your friends car. You never know, just because his seems to run fine doesn't mean something on his car isn't done properly. Using the FSM ensures you're testing the right spots for the right stuff, and that you're following the right procedure.
I'd say "this is just my .02", but that's more like a dollar up there