Sam,
how are the discs? If they've got a pronounced lip at the outer edge, it's worth removing that before fitting new handbrake pads, since the contact point of the handbrake pads on the discs is further out than that of the main pads.
If you fit new handbrake pads to worn discs, the pads will soon be chamfered away at the edges, needing re-adjustment in a short time.
I use a flap disc in an angle grinder, with the car on axle stands and the engine running in 1st gear, but there is probably a safer approach...
Btw, it's the inevitable slight eccentricity of the disc which 'knocks back' the handbrake pads out of contact with the disc.
That's why those little springs are there, to keep the handbrake pads out of contact once they've been pushed back by rotation of the disc.
Also, as a check on correct adjutment of the eccentrics, just look as the position of each operating lever with the handbrake 'on'.
All four levers should sit with similar gaps between them and their stops, usually about 3 or 4 mm.