Blk cloud, check the alternator the way the first couple of posts said to, it should be putting out 13-14.5 volts, if it isn't take it too an alternator shop, or open your wallet wide and take it to Mother Deere
I think the 6000 series had an electric fuel pump right where the line comes out of the fuel tank, that could also be a problem
There is also an inline check valve that they put right under the platform, that could be letting the fuel leak back, and there is also a small diaphram inside the hand or priming pump on top of the fuel filter
I have a bunch of JD tractors, and we do most of the work on them, some of the mechanics they use are great and some not so great, I had problems with a 5625 MFWD that it would lose it's prime when you shut it off, sometimes every time sometimes once in a blue moon, drove me nuts, called the service damager at my local Deere shop, he said there was no check vavle, I said there had to be, so I spent a week or two with Eau De Diesel fuel getting poured on me, because everything was underneith and I had to drain the tank every time to do any work on the system, anyway I jacked with it and finally looked up on www.jdparts.com and saw in the parts catalog where the check valve was, and it has run perfectly since
If it has enough power to start then it has enough power to run the fuel pump and solinoid, IMHO I think you have a fuel delivery problem, and the lower the fuel level in the tank the worse they can be
Actually I think the fuel systems in the newer ones are a PITA