1. Mega
2. Copy / Paste
3. Close your eyes and believe that they do it.
Jokes aside, I've moved from Dropbox to Mega; I don't store private information on cloud services, however, some particular stuff was overwritten some times and deleted the history before closing the account.
Before that, I was suspicious too of Dropbox opening files, so I've planted a canary and never was triggered, so, there's that, unless you're some VIP, I highly doubt they check your files.