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We had the same problems today with droplets on which we used the rebuild feature to upgrade the base image. Newly created droplets were not affected but all Droplets which we upgraded from an older Debian version to Debian 9 were affected. The only workaround we found so far is to rebuild the machine with the Debian 8.9 base image and then upgrade the OS manually:
https://linuxconfig.org/how-to-upgrade-debian-8-jessie-to-debian-9-stretch
I have exactly same problem here. The droplet I have build before (debian 9.0) is working nicely (FRA1 data center), but the droplet I rebuilded today (debian 9.1, FRA1) can’t ‘resolve anything’ using apt-get or aptitude. It’s amazing becouse ping and wget are still working fine to the same addressess. I’m also able to use ssh. So I think it isn’t a network problem, but an issue of apt.
I can work now with the other droplet for couple of days, but I would appreciate any workaround there…