Dhclient not updating /etc/resolv.conf

Mint and other modern distros ship with mdns by default, which wraps the regular public DNS with a local "decentralized" wrapper which enables zeroconf support for your local network. Basically, a local DNS server resolves names in the local network it has discovered, then falls back to the (now proxied) public DNS for public Internet resolution, i.e. for names outside of your local network.

In so many words, your resolv.conf is correct and appropriate for this scenario, and if mdns has problems accessing your ISP's nameserver, you should look inside its configuration - though of course, if you don't care about zeroconf support, disabling mdns (and then probably also Avahi) lets you manage resolv.conf in the traditional fashion.

See also e.g. https://help.ubuntu.com/community/HowToZeroconf