You can fight with a broken sword, although it's different from fighting with a normal one.
Also, maybe Aragorn simply didn't want to carry any other sword but the sword of his ancestors that had special meaning for him. That would explain a lot. Okay, he definitely wielded a different sword in Gondor and Rohan, but I could see him not wanting to use any other sword but Narsil unless it was absolutely necessary. Thus, in the North, where those who could recognise the sword knew who he was anyway and the rest could not identify the legendary weapon, he had little reason (except for it being impractical) not to carry it.
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Like the stars chase the sun, over the glowing hill I will conquer
Blood is running deep, some things never sleep Double Fenris
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