Leeward running stay should be slack, windward taut. Both somewhat taut when running. Never should both be slack.

When tacking/gybing, first hand-tauten leeward stay, bring main to centre, through the wind, release (new) leeward stay, set main, set (new) windward stay.

Your onto it Steve, windward hauled on leeward slacked off. Tight hauled you can generally have them both hauled on depending on the spread of the runners and the sheeting angle. When running you obviously can't have the runner on the side that your boomed out on, and on that point I tie a preventer on the boom because if it gybes it could take out the runner and the top of the mast. And in a beam sea that is likely as the gaff does a bit of swinging about and can drag the boom over.
I don't use them all the time only on a long reach or if I'm pressing the old girl a bit hard. But that depends on how solid your rig is of course. Originally mine came from tangs near the throat halyard but I added extras to the mast head to stop the top mast pumping

Depends on how far back the runners are as well. On older gaff cutters in the UK the runners would finish well forward of the cockpit, and the advantage here was that you could have both runners on upwind and not need to ease them. This meant that you could tighten up the leeward runner on each tack and get some forestay tension for the staysail. A lot of more recent gaff cutters have the runners coming further back, which in theory gives better forestay tension, but often means that the leeward runner has to be slack, which is a pain when tacking as it means easing and tensioning runners during a tack.

woko said..
The mast would pump ( flex back and forth ) as the head sail filled and spilled as they do. And yep it was a bit concerning. You have a tried and tested system Steve so Id bet it's set up to best advantage. Runner tails back to leeward sheet winch is the go

Whoops, runner tails back to windward sheet winch, the leeward ones gunna have the jib sheet on it !

You could consider removing the running backstays and replacing them with fixed shrouds aft of the mast. You'll have to do the calculations to ensure the mast is supported properly by the aft shrouds. A good rigger should be able to do that.
When I built my gaff rigged cutter I could not find a reliable rigger to design the rig so I did it myself. After much research I decided against running backstays. There is a trade off of course - running backstay with smaller diameter shrouds, good mast support well aft versus fixed shroud with larger diameter shroud, stronger mast fittings and stronger chain plates to ensure the mast is well supported but chain plate closer to mast to permit downwind sailing. A fixed shroud will mean a downwind sail has the boom almost against the aft shroud and the boom will not be at 90 degrees to the boat (but it is pretty close). A running backstay will allow the boom to swing further forward. I decided that I prefer the simplicity of no running backstays. I have no regrets and my rig still stands. Note, however, that I have not been caught in a storm to push the absolute limits of my rig.
As a guide: my boat weighs 6.5 tonnes, beam is 3m, hull length is 8m, rig is 1x19 1/4" 316 stainless, mast is stepped on the cabin top, cabin top about 0.7m above deck, mast head is 6.7m above the cabin top, aft shroud is 7.9m long, aft chainplate is 0.8m aft of mast.
Good luck and don't let anyone tell you something cannot be done.