You don't need DoH for that. Just use a VPN and configure it to replace your host's resolver as long as it is up.
Another advantage of using standard UDP-based DNS over a UDP-based VPN is that it can reorder packets in flight, so it should have lower latency than anything TCP-based.
The counterpoint is that traditional DNS has horrendous loss recovery and basically no congestion control and these things definitely benefit DoH at the tail.
QUIC will let us have it both ways (and as QUIC has an HTTP definition, its basically a free upgrade for DoH).
How do I RUN my own server? A few minutes of Googling hasn't revealed any DNS-over-HTTPS server that appears production-ready.