You know, you are right. I always assumed he had a good sword too, but now that I think about it, it isn't said. So in this sense, I do think Peter Jackson was right in the movie to leave the shards of Narsil back in Rivendell, safe and honored, until the time came for the blade to be brought forth in full, re-forged use.

Interesting. Why did Aragorn carry the shards of the sword with him? Could it be that he was never sure when he might have to make his claim to the throne of Gondor? Or was it a more useful weapon against Sauron's minions than a broken sword? For Sauron lost the war and physical form because of that broken sword, and so that piece might have scared enemies of the dark lord....and yet, if Aragorn drew it with anything like frequency, word would have gotten round that the heir of Isildur lived and was (wherever he had been spotted). And that would have gotten Sauron to looking for him specifically...which he did anyway, but not so openly as he would have had to had he known for sure just where Aragorn was, carrying that broken but still powerful sword.