It seems your network advertises unavailable dns server addresses (pointing you at your own local machine) or your machine is misconfigured to use local dns which doesn’t seem to be running. I suspect normal browsing might work as it goes via http proxy that does name resolution for you.
Anyways, I feel the issue is beyond syncthing. Basic things like pinging syncthings dsicovery servers (or even google) should fail, hence you need to sort that out.
Perhaps local dnsmasq is crashing or somrthing like that.