Older (2010) Mac mini high Sierra very slow

Two Options. From Cold Boot immediately hold Command + R to get into Recovery Mode. From there use DU to run First Aid on the drive. Option 2 - download a TRIAL copy of Carbon Copy Cloner. Have an external drive attached and large enough yo handle ALL your Data. Make a Clone of the drive. Once completed, leave the Clone drive attached. Shutdown computer. @ Startup Hold down Option key. Choose the external drive. It will load slowly. Once up and running again use DU ( Disk Utilities ) to run First Aid on your internal drive.

That is a decision only you can make. A 10 year old machine may have seen better days. I have a Mac mini 2011 that has been upgraded to 8 GB RAM and upgraded to 500 GB SSD. It run HS fine but . . can go no further regarding macOS and a single Hardware issue I can not resolve.

The two most likely possibilities are an issue with third party software or a failing hard drive.

First try running Disk Utility First Aid on the drive. Even if First Aid reports everything as fine make sure to expand and examine the details to see if there were any unfixed errors listed especially if the drive is formatted with the new APFS file system. If there are unfixed errors listed, then these can severely impact system performance. Assuming they are not caused by a hardware issue the only way to fix the file system errors is to clone the drive and erase it before restoring the from the clone.

To check the health of the hard drive(s) run DriveDx.

To check for possible software issues run EtreCheck.

Post both reports here using the "Additional Text" icon which looks like a piece of paper.

The MacMini of 2010 is NOT compatible for anything higher than High Sierra - so no. The new machine will probably already have Catalina installed