I'm with you - I've never heard a more beautiful sounding guitar. A HUGE part of that sound is because he was playing a '40s J-50 that was (besides being completely hand built out of better raw materials than are used today) already nearly 20 years old when he recorded those first couple of records. It was very much played.
I do own a modern J-45 which gets me as close as I can afford guitar-wise. You're right in assuming the strings play a role in that sound. What gets it even closer is using medium gauge round core strings. I've experimented with 12 gauge lights and they just don't have the same sound as the 13s. And the round cores absolutely make a difference. Modern hexagonal core strings just don't have the same attack or balance. I understand that companies moved to hex cores because they were faster, therefore cheaper, to wind but as you'll know, cheaper usually isn't better for anyone but the manufacturer.
The good news is that there are companies still making them. Pyramid, DR and DeanMarkley. My money's on the DeanMarkley Formula 82/R. Not overly expensive and they last quite a while. I didn't care for theDRs and have never tried the Pyramids.
If you're talking about the recordings, that's an entirely different story. Columbia studios had an incredible recording room not to mention a hand built valve console and the bestmics bar none (Neumannkm56 is what Dylan is photographed using the most on guitar).
He did play a Martin D-18 on those first couple of records too. Which songs he was playing with which guitar, I don't know. Of course, the largest part of the sound has to do with being BobDylan. Give that guitar to anyone else and it wouldn't sound the same. Have at it!