Strange thing with 5mm LEDs and variable PSU
so I am thinking it's leaking AC or something crazy and that is going
  through me and lighting the LEDs?


I think you are on the right path. More than likely your power supply is a switch-mode type with a floating output. Internally, to reduce DC output switching noise, capacitors are sometimes used from the DC output back to the rectified AC line. This reduces the switching noise that is passed through the internal high-frequency transfomer and, of course, reduces the output switching noise but, a side effect is that there is a capacitive impedance from your AC line to your DC output.

It's harmless to people (although you can feel a little tingle sometimes) but it can drive an AC current of up to a mA or so and this is what is activating your LEDs. The LED rectifies the current and emits a small amount of light.

You can prove this be connecting a DVM from the DC output to ground and measure the voltage. A DC voltage measurement will be close to zero but the open circuit AC voltage may be tens to many tens of volts. Again, it is impedance-limited by the internal noise reduction capacitor and should be regarded as harmless.

Here's a picture of a typical SMPS showing the "rogue" component that is likely causing this effect: -



It's connected between the rectified AC line and the return line of the DC output. Image from this related Q&amp;A.