I have the same vintage Mac Mini with a 1 TB hard disk and had the same problem. Here is what I did and some observations.
(1) On September 25, I performed a Mojave update (10.14.5 to 10.14.6). Afterward, the computer was very slow. It took two minutes for a pull down menu to come down.
(2) I called Apple Support and after some diagnostics, we decided to install Catalina using Option-Command-R without erasing any volumes. After that, the computer would not shut down normally and would reboot after a “panic: watchdog time-out”. What I noticed is that I would be able to shut down normally right after the reboot. However, after some time, something will do something to the system so that if I run “First Aid” using Disk Utility, it will trigger a warning (“Spaceman main free count 218039846 dos not match sum of free counts 218126044”, the numbers change each time). As long as the warning is absent, I can shut down normally.
(3) I called Apple Support on September 30, and we reinstalled Catalina using Command-R. The shut down problem continued and some applications would not respond. At this stage, the computer did not verify an application before opening it. That is supposed to be a Catalina feature.
(4) I did a timemachine full backup on October 11 and called Apple Support on Oct 12, the Tech Support person walked me through the followings:
(a) sign off Apple ID
(b) shut down
(c) Use Command-R to get to MacOS utilities (I am already in 10.15.7)
(d) Use disk utility to erase two volumes (“Macintosh HD data” first, and then “Macintosh HD”)
(e) Install MacOS 10.15.7.
(f) Use timemachine backup to restore data
(5) Previous to erasing the volumes, the disk has 5 volumes with 2 mounted. After the installation, it has 6 volumes with three mounted. There was an additional “Macintosh HD data”. I started seeing the computer verifying some of the applications. However, the shut down problem still persisted.
(6) On October 13, I decided to repeat steps I did on Oct 12, but erase the disk instead of the volumes. This link (https://support.apple.com/en-us/HT208496) explains the difference. I erased the disk and left it in Mac OS Extended (Journaled) format, and let the Catalina installer reformat the disk to APFS . I was back to 5 volumes on the disk, the 2 mounted volumes (“Macintosh HD data” and “Macintosh HD”) and three others (VM (virtual memory?), and 2 “not mounted”).
(7) The normal shut down failed on the evening of Oct 13, but has been working the last 2 nights. Now, it verifies an application before opening it for the first time. I have not seen any “spaceman” warning when running First Aid.
(8) One final thing. When the machine boots up after the installation, there seems to be a lot of house keeping activities. It would be best to leave it alone for an hour of two rather than start opening applications to test things out.
It may be too soon to declare victory. I will let you know.