Given that you want to kill the grass and replace it with mulch between the garden beds I think the best solution which is also natural and chemical free is just to smother all the grass and weeds with thick sheet mulch to give you a blank canvas to work with.
Start by removing large plants and levelling out the ground, collect a heap of cardboard from skip bins out the back of your local shops and spread it out over the area so the edges overlap eliminating any gaps that weeds might try to grow through. Next order it a truck load of course wood chips form your local tree surgeon and spread them out at least 2 inches thick over the whole area. Weeds suppressed and yard mulch all at once.
As a bonus all the cardboard and wood chip will break down over time and add organic matter to the sandy soil.
I say go for it both front and back yard.
