Yeah, this isn't scientific per-se, but the hot climate combined with woolly sheep that need to be sheared for their sakes would lead to an excess of wool to be used in blankets, clothing, headwear, and slings. At least that is the case for shepherds. City folk back then who couldn't raise many sheep, if any, would likely have linen at their disposal - slings being one such disposal. Easier to process flax in a city than to keep a sheep, and easier to maintain, too
As for pouch, whether split or woven: who knows? I sure don't. We can assume their slings were similar to Egyptian slings as stated above, but we can assume many things, and whether they're fact or fiction, none can say. One could guess, as per 1 Samuel 25:29 that the pouch of slings was as the palm of one's hand - the Hebrew word translated as "hollow" or "pocket", "כַּף", means flat and hollowed, usually used to say "hand" in the sense of the hand being open and flat, ready to give or take. So, one could guess that perhaps the typical sling of that era was enclosed, either via weaving or by leather.
Regardless, it's not hard to imagine that slingers throughout time can personalize their slings, so maybe while Abigail is used to closed pouch slings from Nabal's shepherds or whomever else, it might not have been the typical. Just some speculation