Italian260z77

So, first post. Tried searching and couldnt find anything, even tried the all magical, Google, and still came up pretty empty handed. Ive been having an issue recently with my Z dieing while driving under normal conditions. However, I can put it in neutral, and about a 30-40 seconds worth of turning the key gets it fire back up. I have started with the easy and cheap stuff first. I replaced both the positive and negative battery cables and cleaned and checked the ground wire. Then I replaced the in line fuel filter that is before the electronic fuel pump, which was very needed. It was running the best it ever was, I let it sit in my driveway and run for half an hour to make sure all was good. Went out for a drive and about 10 mins in, it died again. My next guess is the fuel pump relay, maybe that is causing my issue? I checked the two relays under the passenger glove box and the bigger one looks to be in bad shape. I am no expert but this looks pretty burnt??Also, I noticed that the previous owner was messing with these because the black w. white stripe wires were spliced together??