After the Battle of the Pelennor Fields, Aragorn draws his sword Anduril and says:

"You shall not be sheathed again until the last battle is fought."
The Lord of the Rings, The Return of the King, Book V, Chapter 9: "The Last Debate"

The date is March 15; the "last battle" he has in mind is presumably the Battle of the Black Gate, which takes place ten days later, on March 25.

What did Aragorn do with Anduril for 10 days? He spent the first two days in his camp outside the gates of Minas Tirith, and the next seven days riding from Minas Tirith to the Black Gate.

We know from his dialogue outside Theoden's hall in Edoras that he is reluctant at best to lay the sword down, even in its sheath and under guard.

"It is not my will to put aside my sword, or to deliver Anduril to the hand of any other man".
-The Lord of the Rings, The Two Towers, Book III, Chapter 6: "The King of the Golden Hall"

He is eventually persuaded by Gandalf to comply with the guard's request to disarm, but only after delivering this warning:

"Here I set it, but I command you not to touch it, nor to permit any other to lay hand upon it. In this elvish sheath dwells the Blade that was Broken and has been made again. Telchar first wrought it in the deeps of time. Death shall come to any man that draws Elendil's sword save Elendil's heir."
-The Lord of the Rings, The Two Towers, Book III, Chapter 6: "The King of the Golden Hall"

In light of this, I can't imagine him leaving it unprotected in his tent. Did he walk around carrying an unsheathed sword for two days, then ride along carrying an unsheathed sword for a week? This seems like it would be dangerous, inconvenient, and fatiguing.

So how did he deal with an unsheathed, presumably razor sharp sword for a week and a half?

Note: here's why carrying an unsheathed sword around is a bad idea:

Aragorn stood before the Black Gate, fire in his eyes, Anduril held firmly in hand, shining forth from his hip. Pippin, alight with excitement, ran to his side, and cried, "Strider! Whither has the Dark Lo----". His voice suddenly ceased, for Aragorn, turning to face the halfling, had smote his neck with an unintended blow; grimly he gazed upon the ruins of Pippin's throat, and with displeasure saw the hobbit's head tumble down the slag pile, spilling from its helm as it came to rest in the scree below.

A tear in his eye, Gandalf spoke: "Ever and anon, even the greatest of men may chance to visit such misfortune upon one whom he loves". Aragorn stood silent, watching until Pippin's headless body, which had remained upright through some unseen force of will, began at length to waver, and finally, fell in a heap at Aragorn's feet. "I did not wish this thing, Peregrin Took, Ernil i Pheriannath, son of Paladin of the Shire. Truly, I tell thee, though I fear it be of no comfort now - My bad, dog".