Consequences of not servicing a vintage Rolex?

[fuzzyfrank]

Hi all! I recently acquired my first Rolex! It's a vintage 1603 DateJust. I've noticed that the seconds hand between about ~2 and ~6 can get a little stutter-y, more-so than I expect a low beatrate movement to look like. I'm planning on taking it to a local specialist in my area this weekend to get it looked at and discuss servicing (the jubilee band also needs tightening).

In the meantime, do you think there's any issue with me wearing it? I'm sure it just needs some lubrication or something, but I don't want to destroy any of the parts by having it running and scraping together. In the scheme of the watch's life, I suppose a week isn't much, but I'm still nervous.