Did Aragorn really walk around the Wild with just a broken sword for protection?

[Morwen]

I think it likely that the shards of Narsil usually remained in Rivendell and did not accompany Aragorn in during all his Third Age exploits. Practically speaking, a broken sword, as Aragorn himself recognises, is of no real use.

He drew out his sword, and they saw that the blade was indeed broken a foot below the hilt. "Not much use is it, Sam?" said Strider. "But the time is near when it shall be forged anew."

Strider, FotR

Reading the last line I get the impression that Strider had only recently taken to carrying the sword around, as he senses his destiny approaching. Perhaps he was carrying it on that particular occasion as a talisman of sorts, taking the sword that separated Sauron from his Ring as he went in search of the Ringbearer.