So Other
than seeing what :
ip r     says
and say ask the end point for a dns resolution

when it connects ($?)  and then set that address  into /etc/resolv.conf
approx...

OR
exploit the up down calls  or manual types in /etc/interfaces
and fire up a script that does it for you.
UMmmm
Roughly

So yeah these are non standard and could be some place else
These would write what YOU want in that file and maybe restore something else on interface down.
The options are optional and from my memory set in the calling environment in any case.

You could do similar when ever the connman daemon is restarted.

Whatever is easier or actually a sane solution for you.

Eventually, I guess you will need to tell routing where to go.

OR try dhclient yourInterface and hope

I have a rather over featured examples of that localscript "breakage"  here I can sanitise and post if needs be.

Use your own NIC names and addressing of course

I've never previously used "connman" (connmand).
No manpage AFAICT, and all the online tutorials I found cover only dhcp (vs static).
After hand-editing the eth0 config to specify "dns-nameservers xx.xx.xx.xx" line in /etc/network/interfaces, I also tried

`connman -n --nodnsproxy`

and eventually found my way to editing /lib/...connman/eth_looooong_idstring/settings

I've stopped/restarted the network service
I tried clearing the settings via `ip route del default`

don't know what else to try. Can I buy a clue?