To copy-paste, keep both instances of ArcGIS Pro opened (old and new)..
@ajwright
You can check backup folder , where your project is saved, maybe you can find something there:

What is the version of ArcGIS Pro in which this map was created? What is the version now of ArcGIS Pro? Have you recently upgraded or downgraded the application?
Have you installed/updated any add-ins recently? If yes, try uninstalling the add-in.
Do you have ArcGIS Data Interpretability installed, if yes, and you have no issues uninstalling it, YOU CAN GIVE IT A TRY..
If nothing else works, I would suggest you to reach out to Esri Tech support, there might be a chance that they can recover the map from your aprx file....I've seen cases in past, where map has been recovered by Esri tech Support...
Is there a way I can access the layers and paste them to a new map? Your feature classes/shapefiles must be safe at there respective locations...however, I don't think there is any other way to recover the layer (with configurations such as symbology, labels etc.)...