Well, it's a bit difficult to give remote recommendation for a system in this state.
(Ideally I want to login to it by myself ...)

It would be good to know how many of the installed packages are still packages from impish, and how many already from jammy.

One, may try to move forward to jammy (and forcing it),
but I think it's better to be careful and go one step back first of all.

So I would replace all occurrences of jammy in /etc/apt/sources.list back to impish
and do an 'apt clean' and 'apt update'.

Now I provoked a /similar/ situation like yours by manually downloading the jammy package and force installing it:

$ wget http://launchpadlibrarian.net/592653998/libpam-modules-bin_1.4.0-11ubuntu2_s390x.deb 

$ sudo dpkg -i --force-depends ./libpam-modules-bin_1.4.0-11ubuntu2_s390x.deb
(Reading database ... 83842 files and directories currently installed.)
Preparing to unpack .../libpam-modules-bin_1.4.0-11ubuntu2_s390x.deb ...
Unpacking libpam-modules-bin (1.4.0-11ubuntu2) over (1.3.1-5ubuntu11) ...
Setting up libpam-modules-bin (1.4.0-11ubuntu2) ...
Processing triggers for man-db (2.9.4-2) ...

After that apt full-upgrade shows the same error like on your system:

$ sudo apt full-upgrade
Reading package lists... Done
Building dependency tree... Done
Reading state information... Done
You might want to run 'apt --fix-broken install' to correct these.
The following packages have unmet dependencies:
 libpam-modules : PreDepends: libpam-modules-bin (= 1.3.1-5ubuntu11) but 1.4.0-11ubuntu2 is installed
                  Recommends: update-motd but it is not installed
E: Unmet dependencies. Try 'apt --fix-broken install' with no packages (or specify a solution).

Now - since we want to go back to impish - the opposite can be done to solve the situation, means forcing the installation of the impish package:

$ wget http://launchpadlibrarian.net/558628390/libpam-modules-bin_1.3.1-5ubuntu11_s390x.deb

$ sudo dpkg -i --force-depends ./libpam-modules-bin_1.3.1-5ubuntu11_s390x.deb
dpkg: warning: downgrading libpam-modules-bin from 1.4.0-11ubuntu2 to 1.3.1-5ubuntu11
(Reading database ... 83840 files and directories currently installed.)
Preparing to unpack .../libpam-modules-bin_1.3.1-5ubuntu11_s390x.deb ...
Unpacking libpam-modules-bin (1.3.1-5ubuntu11) over (1.4.0-11ubuntu2) ...
Setting up libpam-modules-bin (1.3.1-5ubuntu11) ...
Processing triggers for man-db (2.9.4-2) ...

After that apt is happy again:

$ sudo apt full-upgrade
Reading package lists... Done
Building dependency tree... Done
Reading state information... Done
Calculating upgrade... Done
The following packages were automatically installed and are no longer required:
  apport-symptoms eatmydata libavahi-core7 libdaemon0 libeatmydata1
  libregexp-assemble-perl libwrap0 python3-debconf python3-jinja2
  python3-json-pointer python3-jsonpatch python3-jsonschema python3-markupsafe
  python3-pyrsistent python3-systemd squashfs-tools
Use 'sudo apt autoremove' to remove them.
0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.

(Please notice that forcing packages is usually a bad idea and can break the packaging system, and should only be done with caution and in rare corner cases!)

And it could be that more packages are affected and in such a 'limbo' state,
so further steps like this might be needed.

Once apt doesn't report further issues, and a
sudo apt update && sudo apt full-upgrade
runs fine and you ideally restarted the system afterwards,
it might be the time for a new "do-release-upgrade' ideally in a screen session,
in case the connection breaks and to save the output).

(But please think about a backup before doing all this ....)