For my braies I used naturally coloured, finely woven real linen. After a couple of washes this material is as soft as cotton, but - even given the amount of fabric involved - the cost of linen shouldn't be seen as excessive: linen braies should last years.
Cutting the cloth, stitching the seams and finishing the hems takes only a few hours when sewing by hand.
The drawstring in my braies is simply a long piece of tablet-woven braid, made into a circular section by passing the weft the same direction through the shed during manufacture. No doubt a thin leather belt or a long fingerloop braid would also be appropriate.