This problem is fixed.

The cause of the problem has to do with the PackageContents.xml of a _different_ add-in in my computer contains an "inappropriate" version number.  It was like this in that PackageContents.xml file:

<RuntimeRequirements ... SeriesMin="R19" />

R19 is for AutoCAD version 2013 to 2014.  My computer doesn't have that version of AutoCAD. 

That add-in also needs to use Fabrication API.  I had been focusing on the add-in that I am working on, and I didn't realize that the problem was coming from a different add-in.

With that wrong version number in PackageContents.xml in that add-in, when I launched CADmep, seem like CADmep chose to launch the oldest version of FabricationAPI.dll in my computer and that happens to be version 2019.  This explains the reason why the 2019 version of FabricationAPI.dll has been pre-loaded when I launch CADmep 2023.  And that somehow prevents my program from loading 2023 version of FabricationAPI.dll.  This explains the reason why I had a problem loading 2023 version of FabricationAPI.dll in CADmep 2023.

As for the reason why I cannot get CADmep 2021 to load "any" version of FabricationAPI.dll, I still don't know.  Nevertheless, the problem in CADmep 2021 is also fixed.

The fix is to enter the correct version of AutoCAD in PackageContents.xml in that add-in, like this:

<RuntimeRequirements ... SeriesMin="R23" SeriesMax="R23"/>

R23 is the version number for AutoCAD 2019.

Now, I find that CADmep can pre-load the correct version of FabricationAPI.dll or CADmep can allow my add-in to load the correct version of FabricationAPI.dll.  And the add-in works properly in multiple versions of CADmep.

Hope this helps someone.

JC_BL