Can I double wrap old pasul tzitzus and throw them out instead of giving them in to sheimos?
If the tzitzis are indeed pasul, they are considered tashmishei mitzvah, (an item that was used for a mitzvah, but can no longer be used for the mitzvah). Tashmishei mitzvah should not be discarded in a degrading way, like to throw in the garbage, which is disrespectful to them. They may be put into a single plastic bag, (preferably opaque) closed and then placed in the garbage. This way it doesn’t look like we are treating them with disrespect. This is the basic halacha, it is however preferable to put them with the shaimos
O:CH 21-1, M:B Ben ish Chai 1 Lech Lecha 19, Kaf Hachaim OC:H 21-2, Shevet Hakehasi 4-10, (that this is also for Sefardim), Pischei Tshuvos 21-1.