Stop worrying with resolv.conf and /etc/network.
Install
root@debian:~# apt install systemd-resolved
# now or after dhcp
root@debian:~# systemctl enable systemd-resolved && systemctl start systemd-resolved
Move the old interfaces file
root@debian:~# mv /etc/network/interfaces /etc/network/interfaces.save
Minimal /etc/systemd/network/10-eth0.network
[Match]
Name=e*
Type=ether
[Network]
DHCP=ipv4
# Use this if you want to force DHCP-Registration with
# the MAC-Address instead of duid (DHCP Unique Identifier)
# This way you can pre-assign them in your DHCPD.
#[DHCPv4]
#ClientIdentifier=mac
Check the actual device in dmesg
e1000e 0000:00:19.0 eno0: renamed from eth0
r8169 0000:01:00.0 enp1s0: renamed from eth0
So notice they can be different then match this in Name=eno0 or Name=enp1s0 or just stay with e*.
Do not try e*0* or so, just use the device name.
Enable the service and reboot, cross fingers!
root@debian:~# systemctl enable systemd-networkd