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How do you update applications to judges who do not have an email on OSCAR? Is it ok to send the update snail mail even if they do not take paper apps?

Or should I send the update to an email I find for the judge on the court's website?

Also, what if the judge says "do not contact chambers"? Is it still appropriate to send an update?

I would just send your resume via snail mail. In your cover letter, explain that you previously submitted your application on OSCAR, but you needed to amend your resume for whatever reason. You can also point out that you are mailing it because of the court's preference that you do not contact them, but you are happy to email a copy if requested.