I have to upgrade from 17.3 as I need the systemctl to install Virtual Shield. I used to be able to save all my installed packages using dpkg redirected to a file, backup the file and then install on the new system using dpkg and the back up list. I can't see that the install from the list is still possible - of course I might be remembering wrongly. The backup tool doesn't seem to work. On one PC I get the list but it is not saved to the chosen directory. On the other PC it doesn't find and files - both have almost the identical programs added. I've also noticed when I've packed up previously that the backup tool didn't list all my installed programs, which dpkg-l did.

I would also like to save my PPA sources to be able to add them again. I've copied the files from the etc/apt directory so I can add all by hand. I've installed Aptik but can't install the GUI - package not located. I can't find Aptik in the Menu either. The repository has been added and update run.

17.3 of what Linux distribution?

linux mint rosa 17.3

As part of your new install you can select the partition to use. If I remember correctly you can also change the size of current partition(s) before creating the partition to use. If that is the case, you can use the approach of resizing your existing partition to near minimum and create a new partition in the new unallocated space if it is adequate. Then install to the newly created partition. Do this as part of installation, not separately. Changing the boot partition in any way can prevent booting which requires the reinstallation of grub which is a royal pain, so do it when grub is going to be installed anyway, i.e. during system installation. You will now be in the position you can mount your old partition in your new system and have access to all files until you are sure you don't need them anymore. As a matter of fact you will probably be able to boot into your old system if you need to because the os-prober will find it when setting up grub for your new system.

You might also look at something like rsnapshot (that uses rsync) to do your backup, or use rsync directly to back up the directories that contain the files you need to keep. But be sure keeping the files is all that is necessary. If you keep them and later find out it was not enough it will be too late if you have deleted everything old. That's why I recommend the first approach if it is available, i.e. you can create enough free space. A new installation doesn't require that much space and you can always add more later.

Hard drives are not that expensive these days. If free space is a problem is adding another drive a possibility?

Using 2 separate machines should be straight-forward. What I have found useful is to have a separate /home partition - make sure you copy that over first, and during the install choose "something else" for the partitioning and assign /home to not format. The /home has all the customised config settings for each package, so minimises post-processing.
Do the install and use dpkg to generate and repopulate the installed packages as you mentioned - there must be a million posts on this on-line. People are always warning about package version conflicts,but I've rarely run into any. As it happens I generally take the opportunity to purge the list, and only install what I actually want - still leaves cruft in /home/.config or whatever that you can clean up at your leisure.

I always do ALL backups on a separate USB disk, one of my super strong points when I gave courses on security. I've saved my packages using dpkg get-selections and set up my January 2020 folder on my backup disk. My problem now is that I want to use Aptik which seems to simplify the process. But the GUI (aptik-gtk)isn't found in the package list. Using the from the command line would defeat the purpose of simplicity. The PPA is:
deb http://ppa.launchpad.net/teejee2008/ppa/ubuntu trusty main
deb-src http://ppa.launchpad.net/teejee2008/ppa/ubuntu trusty main
Does anyone have any ideas on this?