You don’t need a special bag to wash tzitis. You can put the strings in a knee-high and then twist the opening into a knot. If the strings are dirty pretreat them first like you would anything else. (Don’t leave strong pretreatment on too long though.)
If all else fails you can buy replacement strings in many sforim stores, which you need to know how to tie.
Knee highs should be placed in the mesh bags to avoid any sortta ripping… Just like the tzistzis… Unless its an old knee high that you don’t care about… I wouldnt really recommend it!!
It is 100% ossur to even think about putting tzitzis (a MALE garment) into a f-male article of clothing, even if it is to protect it. Your not even supposed to type the word **male (the word that indicates a person that is not male!!) What is wrong with you people. THIS IS WHAT THE INTERNET DOES TO NESHOMAHS!!!