Wearing Tzitzit Before Washing One's Hands

It is brought in the Shulchan Aruch, Siman 8, Se’if 1: to wear tzitzis, etc., and the Mishnah Berurah writes: immediately after washing one’s hands, so as not to walk four amos without tzitzis.I have seen in the Acharonim two reasons why one should not wear tzitzis before washing:Because it is forbidden to touch clothing before washing one’s hands.Since one must recite the blessing before wearing the tzitzis, and cannot do so because he has not yet washed his hands. My question: In my house, I cannot wash my hands within four amos of my bed. Should I therefore put on tzitzis first so as not to walk four amos without them, or not— so as not to forfeit the two reasons for not wearing tzitzis before washing one’s hands in the morning?! 

Shalom uVracha,

If one will not be washing his hands immediately, he should first put on the tzitzis, thereby fulfilling the mitzvah of tzitzis right away.