Before someone else jumps on you from a height, get a signature sorted out mate.
E53 covers a lot of models. I'm sure that makes no difference to your 'current' (see what I did there? current, no power..

) situation as they will most likely be generic back there. However it will do you favours in the future.
And when I say future, I mean you will spend a lot of time here seeking advice on this motor!
Unfortunately I can't offer any more advice because I've not come across this issue myself. But yes a continuity test would be favourite IF you can find where to test from! Good luck, and let us know how you get on..
__________________
Present:
01 X5 4.4i
07 Yam R1
Past:
89 R32 Skyline various mods, running 442 BHP
00 Merc CLK55 AMG
01 E39 M5 Haywood and Scott silencers
02 E46 M3 Supersprint Cats, Eibach Lowered, Thorney Motorsport Remap, running 368 BHP