Is it just dropping the DNS or is it dropping the wifi entirely? I've been having an issue in my district where chromebooks will just lose their wifi connection out of nowhere. Most of the time they can just go into the wifi menu, click on the correct one, and get back online but occasionally it prompts for a password like it's forgotten it. Best guess is if it loses the wifi x times it prompts for the password.
Similarly to you though a reboot tends to fix that last issue. In my case I can't get a solid bead on why it's happening and I'm half hoping our pending WAP refresh helps. It's not just a matter of chromebooks going to sleep and disconnecting from wifi I've gotten reports of being in active use and just dropping.