I have a case 580 SE backhoe that after sitting all winter(Michigan) the ignition switch froze up(key would not turn). i replaced the ignition switch and problem solved! Now after starting/using 1/2 dozen times the engine will only start and keep running if you hold the key in the start position, as soon as you let it go to run it shuts off. i'm sure its a simple fix, just need to know where to look. Also need to pull the steering wheel to replace the steering wheel bearing, does that require a special puller or can i just use a 3 armed puller?
Im not sure on your machine but does the injector pump have a 3 wire solenoid valve on it? If so, check your fuse panel and make sure all your fuses are good. The symptoms your describing sounds very familiar with those type of solenoids. There is a ignition side of the coil and a start pull coil. Usually when that happens the ignition side either doesn't have power or that coil is fried.
I've owned a great many older machines in my life. Most common problem: bad connections in the wiring. My Case backhoes are tenfold the worst. Usually, a new wiring harness costs thousands. In most cases we get hands on a schematic, and replace the offending wire as needed. I'm an electrician, and it drives me CRAZY! For a non electrician to figure it out, I imagine would be infuriating! Often it comes down to a directive from a control failing to reach the instructed device. In most cases, a connector mid wire corrodes. Been through it too many times on 580K.
Check inj. pump solenoid to see if you have voltage there all the time with key on. If not then trace back to key switch to check for bad connection or broken wire. Also did you replace key switch with Case switch?