As best I understand it, running a private caching DNS server is the only guaranteed increase in privacy for DNS. That server still has to reach out to the net the first time a request is made, but will resolve all subsequent requests locally. DNSSEC to a privacy respecting DNS provider like quad9 at 149.112.112.112 from your local DNS server. Mayhaps the best you could do for a roaming device like a phone is to run a decent VPN with an option to prevent DNS leaks.
As best I understand it, running a private caching DNS server is the only guaranteed increase in privacy for DNS. That server still has to reach out to the net the first time a request is made, but will resolve all subsequent requests locally. DNSSEC to a privacy respecting DNS provider like quad9 at 149.112.112.112 from your local DNS server. Mayhaps the best you could do for a roaming device like a phone is to run a decent VPN with an option to prevent DNS leaks.