I'm wondering if it's a sticking fuel shutoff solenoid. I wonder if a shot of diesel fuel conditioner or perhaps a bit of Seafoam or the like in the fuel system might clean things up a bit. Cheap way to try and solve your problem.
Either way, keep us in the loop!
BTW, welcome to the forum.

 

I agree with Pogobill's assessment of the fuel shutoff valve.

To me, a clicking solenoid is an indicator that the battery is failing. Or, you may have dirty connections. But if the battery is old, it may not be in "good shape" as you state.

 

Since you had the starter off you obviously know where the start signal wire fits onto the solenoid. If you remove this wire, replace it with a jumper wire and engage the starter with direct battery voltage(ignition OFF, trans in neutral), how does the starter perform? Does it engage properly, crank the engine as it should, and disengage when power is removed? This quick check will eliminate the starter as a possible problem, bringing you back to the start signal circuit. Quite often tractors with a complex safety start system end up with low voltage at this point when the key switch is turned to the start position resulting in issues with starter operation. Key switch contacts are a known culprit, as are the related safety switches, relays, etc involved.
As for the slow shut down, you might locate the fuel stop solenoid on/near the injection pump. There should be a single wire pushed onto a spade terminal. This is the run/stop signal. Once located, start the engine, and maybe with a pair of needle nose pliers(if you can't reach it with your hand), pull the wire off. See if the engine dies as quickly as expected, or does it still take the extra few seconds to finally die. This may help identify a mechanical vs an electrical problem. I doubt fuel condition is a factor here, as no fuel ever comes in contact with the solenoid or the linkage it operates. All this is bathed in engine oil, not fuel.