I'm trying to rearrange the docked icons in Zorin 16.3 core by left clicking and dragging. It doesn't work and it just opens the app once I let go of the left mouse button. In order to rearrange the icons, I have to undock them and add them back in the order that I want.
Your Profile says you are on Zorin OS Core, but what you describe sounds like the panel on Zorin OS Lite.
On Zorin OS Lite, right click the panel, move to the Panel > (With the right arrow) and follow the arrow to open the popover menu that offers Panel Preferences
Click that and it will open a Popup Window with the Panel settings. Click on the third tab Items
You can rearrange all panel items in this window by using the Arrow Icons on the right hand side to select an item and use the arrows to move it up or down.