First I suggest cloudflare dns over google ip is 1.1.1.1 for ipv4. They dont log queries (at least not tied to requesting ip). Plus in my opinion its faster and they support dns over tls.
Second try to use a dns server like unbound that supports a setting called "serve expired" as your caching server, basically it will keep the most recently fetched record response cached forever until a new query is requested, and then it will only do a background update not making the live query wait for it. Its a game changer in this era of very low ttl dns records.
If you have no lan caching server, then get one set up.