Okay, now I'll show my age. LOL. I once owned a stuffed animal (it was a hand puppet too) that was stuffed with sawdust. Yep back in the dark ages when dinosaurs roamed.
For an idea of timing, I'm about to turn 50 and I'm pretty sure, the toy - a leopard hand puppet was an antique even then that had possibly belonged to my mother.
So yeah, back in the old days, before we used to have fiberfill, many stuffed animals were filled with sawdust. Cotton, shredded rags, shredded paper and straw were the other popular options.
I could be wrong, but I seem to recall that in the Velveteen Rabbit, it speaks of the rabbit (or perhaps one of the other toys) leaking sawdust because it had been well loved.
So sawdust blood is not much of a stretch, and if Martin is my age or older, not terribly surprising that he'd recall it.
I also recall that when I was a child, sawdust was regularly used to clean up oil spills and such. (Of course grandpa was a carpenter, so we had plenty of the stuff in stock.)
Incidentally, when I studied as an "auto bodyman" I found that the stuff we used to clean up oil spills was basically the same clay as kitty litter without the scent, and WAY WAY less expensive. You could buy a 100 lb bag of the stuff for about $5.
Anyway, for whatever reason, you could often buy sawdust that had been dyed red for use in cleaning up spills. I have no idea what the purpose behind the red dye was. It wasn't paint, but probably the same kind of dye that's used in food coloring. So very simple to translate red sawdust into puppet blood and probably something that was common in the past.
Oh yeah, GRRM is born in 48, 4 years after my Mom. They definitely still had sawdust filled stuffies/puppets back then. Probably not a matter of great research, but one of simple memory.
That doesn't stop it from being an evocative phrase, but maybe makes it less of a mystery. Some of us just know old tech, not because of research but because we're well...old.
ETA: The red thing. My hubby says he can't recall the particular red chemical, but he seems to recall that the chemical used was a fire-retardant included to keep the cleanup sawdust from spontaneously combusting.