Sounds like a possible wiring cross up John.  YES, the ground wires going to the injectors from the EC-2 do control the on/off of the injectors.  The injectors has 12 Volts applied to one of its two pins, the other pin runs to a injector grounding pin on the EC2 (don’t have my diagram right handy).  When the EC2 is not running but there is power to the injectors there will be 12 volts on both sides of the injector.  This is because in order for the EC2 to open the injectors it has to complete the circuit to ground.  When the EC2 does ground its end of the injectors lead, then current flows through the injector, yanking it open and fuel flows.  So if there is for some reason power to the injectors and for ANY reason one of the injector wires is unintentionally grounded that injector will lock full open and if pumps are running will quickly flood your engine.  The EC2 does not have to be running for this to happen if your wiring has a short to ground on one (or more) of the wires running from the injectors to the EC2.

So check it out.  It sounds like when you put the fuse in then you are providing the power to the injectors which appear to have their EC2 side continuously grounded (at least one of them).  The EC2 when operating will periodically ground the injectors to open them – but, will not ground them when turned off (or rather SHOULD NOT).  So check with a volt ohm meter between each of your injector leads going to the EC2 (I presume there are 4).  Put one lead of the ohm meter  on each of the injectors wires going to the EC2 and the other lead to a known electrical ground with the ohm meter scale (if its not automatic) set say on 100 Ohms or thereabouts.  , I am not certain what the ohm reading should be – but it should not be very low.  if its reading less than 50 ohms there is probably something shorted, if the reading is near 1 ohm then its almost certain you have a short to ground..    In other words if you have anything indicating that an injector lead to the EC2 is grounded then that is the problem.

Ed