Did you ever resolve this issue? I am having what seems to be the same problem. At first I thought it was just the fuel pump because I did see it leaking when I could get it running. I changed the pump and mowed for several hours. I suddenly stopped and I found that the only way to start it was to push the choke lever forward and even at that I had to nurse it back to the garage. I was beginning to wonder if I got a bad pump.
I figured that this problem had to be fuel related, so I took off the fuel tank and found a cracked grommet where the pickup tube comes out of the tank and into the fuel line. I thought it was sucking air into the fuel line. I changed the grommet and all fuel lines... problem persisted. (Fuel tank was clean, filter was changed.)
Then I changed the carburetor. I had taken it apart so many times I thought maybe it was damaged. No good... problem came back.
Then out of desperation, I changed the fuel pump. It was the old style pump that was at risk of causing fires. The new style pump has a line that would route fuel safely away from the muffler in case the pump ever leaks. I'm happier with the new pump, but again this was not the problem.
I went to my local John Deere and spoke to the service manager. He told me the problem is common and that I needed to change the Time Delay Module. "All the symptoms fit", he said. I went to the parts counter and made absolutely sure this Time Delay Module was applicable to my 1998 FD590V engine. They assured me it was. Unfortunately, the dealer was mistaken. The FD590V engine does NOT have a Time Delay Module. This was pretty obvious if he had bothered to read the notes on the exploded parts view.
So now what? The entire fuel system from carb to tank has been replaced and it's still stalling out after an hour or so on hot days. I called the dealer back and their new theory is the Engine Control Module... for $300!! I did some Googling and found out this module used to be available after-market as recently as two years ago for about $80, but not anymore. Even on eBay, they cost at least $250. Some people had reported success by dismantling this module, checking for broken wires, etc. I went to remove mine and discovered a loose ground wire fastened to the top mounting bolt. At some point in the past, this fastener had snapped off so I had to drill it out and re-tap the threads, and apparently this eventually failed. I re-drilled and tapped for a larger diameter bolt and fixed the ground wire. No good... problem came back.
So back to the module itself. I used a Dremel to take the back off the case and get the circuit board out. There was some minor corrosion from water, but all wiring and components appeared intact. I realize that internal electrical issues cannot be seen, however, broken wires, corrosion, and heat damage can be. I saw nothing obvious that might explain failure. I cleaned and reassembled the module using liberal amounts of clear silicone sealant.
Given the loose ground wire, I figured the next step would be to inspect every wire and connection from the spark plugs to the battery. I replaced both spark plugs, removed both ignition coils and another mounting bolt snapped off. Luckily, I was able to get a vise-grip on the stud and back it out without damaging the hole or threads. I took a wire brush on the Dremel and cleaned all mating surfaces and electrical connections. The conductors can be removed from the electrical connectors by slipping a tiny screwdriver inside to release the holding tab, which makes cleaning much easier. At this point, I took apart every cable connector on the tractor after first taking photos of each ensuring the wires go back into the same positions.
Now things get interesting. The problem does not fully go away but it becomes MUCH LESS frequent. When the problem used to start happening, it would then die out every minute or so making mowing nearly impossible. Now it might die out twice in two hours. I'm on the right track...
I focused more attention on the two connectors on the main starting board... don't remember what it's called but it's mounted on the right side up near the battery and the key switch is connected to it. These electrical connectors were very corroded and one of the main capacitors had been changed years ago. I thoroughly cleaned this board and every one of its electrical connectors. I was able to mow the yard a few times before the problem came back... when the problem occurred... I wiggled this connector and the problem went away.
At some point I'll replace this starting board and/or clean the electrical connections again, but for now, the problem is almost completely gone.
I can't say that your issues are being caused by the same problem as mine... however, the lesson here is that these symptoms of dying out, using the choke to keep the engine running, and then running normal again... can be starving for fuel AND/OR intermittent electrical... so basically anything. Good luck.