If I want to say it politely, what should I say instead of "dig through"?
Instead of saying "dig through" my inbox, you can use the expressions "look through," "check," or "inspect." However, note that using "don't really" as a way of expressing hesitance or reluctance is also quite casual, so the entire sentence would need to be reworked to become more appropriate for professional settings. Ex: I would rather not check my entire inbox. Ex: I'd prefer to not look through my entire inbox.