Even with the pain, highly doubtful to be osteoarthritis if that's what you mean. The cartilage and all that jazz in there is probably still fine.
That said, overuse injuries of all kinds can pop up just about anytime, quickly, and be next to impossible to get better without a total break from the inciting activity (and even then, maybe never).
Not as medical advice, but I highly advise all physicians but especially med students and residents (before having excellent own occupation disability insurance) to essentially drop everything and pull out all the stops the minute something like it rears it head. Something like this getting out of control in med school or residency can easily end your career before it even starts. Don't think it can't.
Start taking steps to get this under control. If that means more time to start mastering Dragon or even paying out of pocket for it, do it. Look into getting a more friendly keyboard, and lug it around and plug it in as you need to for whatever you're gonna have to do on a keyboard anyway.
Don't ever do less than the minimum required of your notes by your program or from a medicolegal perspective, that said, learn what is and work toward that absolute bare minimum.
Ditto making love to your EHR system and learn it inside and out, improve your templates and dotphrases. If it takes you time after work, do it. If you have to go in on your day off to an appt with one of the technicians to get your templates and dot phrases even better, do it.
I can't give you medical advice or tell you not to see a doc about this. I personally would start reading everything I can online about this problem and trying solutions. I would self treat with ice and perhaps NSAIDs in addition to looking for PT type solutions online.
Your hands may just toughen up with time, or you could just develop some sort of poorly understood chronic pain condition and lose not only your livelihood, end up in poverty, but also the ability to write more than a paragraph at a time or braid your child's hair or make your baby the baby blanket you always dreamed of, or even to masturbate. Do what you can now if there is any way to prevent any of the latter. Trust me on this.