Glad you asked this Sarah Lou as the same question pops up when talking with other gardeners.
The answer is YES, sow and plant!! straightaway!!
Your cardboard will actually rot quite quickly under the damp manure and acts to slow down, rather than totally stop, the pushing up of couch grass and whatever other perennial weeds you have.
However it is helping, along with your six inch mulch of compost and manure, to weaken all weed roots and cause most of them to expire in the next few weeks or months. A few shoots will probably emerge between your healthily growing vegetables, and you can slip a trowel underneath them while admiring your food plants.
I have sown carrots in a bed of newly filled compost, six inches on top of green grass, buttercup and dandelion. The carrots grew brilliantly and none of the grass or weeds grew through, so they simply decomposed and turned to bare soil, under the compost, while the carrots (and lettuce, spinach, beetroot, peas) all grew happily on top. I am pretty sure that they were rooting into the soil around decomposing turf and weeds, for moisture at first, then nutrients after two or three months as decomposition was finishing.
And I hope your back recovers, maybe check out ‘inversion tables’ a great help to my back.