If I understand you correctly, I think you can manually create symlinks to the HDD. For example, if you want libreoffice suite to be on HDD, you would go into where the libreoffice suite is located on the SSD, move it to some location on the HDD, and create a symlink in its place pointing to the new location. This way, the data is stored on the HDD but all programs still find its way there.
I would also recommend upgrading your HDD to a SATA SSD. But of course, that costs money so, up to you.
That kind of looks like how a machine would look if the RAM is loose in the socket or otherwise have a bad contact. And since you mention that it's random and sometimes there are no vertical lines, that'd be my guess rather than anything to do with graphical software. Is the RAM on your machine soldered on or is it detachable? If it's the latter, I recommend taking the RAM out and reseating it to see if things improve (try this a couple of times, just to be sure the RAM is properly seated). If the RAM is soldered on then it might have something to do with out of memory. Does the system slow down a couple of seconds before freezing entirely? If it does then it's likely it's a OOM thing.
Oh, and for good measure, can you get to the tty (ctrl + alt + f2) when the system freezes? If you can then it's probably a graphical thing and you can try restarting those processes. If the system is in a complete freeze then I'd say that's another point for the OOM hypothesis.