With inputs shorted I cannot hear anything with headphones on the S.E.X. with the volume at max - silent.
This means your amp is not the issue.
Remove the shorting plugs (so nothing is connected) and I can hear what sounds like a (very faint) 120hz hum.
Try turning the volume down a bit on the SEX amp, that might help. This is the SEX amp picking up ambient grunge in your environment.
I hear this same hum when the S.E.X. is connected to my preamp, a NAD 1300 that I restored. The cable is an audioquest evergreen cable 1m long.
Same recommendation as above.
I did not hear any hum from the solid state amp (a NAD 902) which the S.E.X. has replaced.
Solid state amps have dramatically lower input impedance and a fair number of precautions taken to avoid amplifying environmental noise. It may be that a 10K resistor across each RCA jack at the input of the SEX amp would replicate this. It may also be the case that you need to touch a wire between the chassis of your SEX amp and the chassis (grounding post) of the NAD to see if there's some kind of grounding issue.
The NADs do not have a ground connector on their plugs - so could be contributing to the issue - I think I read a post on this forum that I could connect a ground wire to a screw on the S.E.X. back to the NAD preamp and that may resolve it. Just want to be sure this is an ok thing to do
Well, as the title of your post suggests, you know what to do!
- don't want to create an electrical problem that blows up my new amp 
You'll have to do something a lot stranger to cause that kind of damage.
; I suspect this could be background noise from some transformer in the topology being picked up by the cable too.
Does the noise get worse if you have the cables plugged into the SEX but not connected on the other end? (Or try temporarily wrapping a 10K resistor from center pin to shell on each free end to provide a reasonable approximation of being plugged in)
