I don't fully understand the problem you're describing, but if you have Pro 3.0, you can use the Copy Network Analysis Layer geoprocessing tool to make a copy of the OD Cost Matrix layer and all its analysis data. The tool doesn't directly allow you to set the output geodatabase location, but you can control the output location by setting the Current Workspace environment.
On the Analysis tab, click the Environments button.
In the Environments dialog, set the Current Workspace environment to the geodatabase of your choice.
When you run Copy Network Analysis Layer, the analysis data of the original layer will be copied into this geodatabase, and a copy of the layer will be added to the map.