Yes! Check out the ArcGIS Assistant, as it can make this possible. The addition of the Map makes this a bit more complicated, but still doable.
Open your dashboard and then choose "save as" to create your new copy.
Go into the ArcGIS Assistant and open up the copy dashboard's JSON for editing. Look for the mapwidget item that refers to your map. You should be able to change the itemID being referenced.
Then go through the JSON looking for dataset arrays.
This is where you'll update the itemId of the layer datasources and their IDs.
It will likely take some trial and error, but if you know which layers are being replaced and have their itemIDs at hand, you can get your layers swapped out without needing to reconfigure all your widgets.
- Josh Carlson
Kendall County GIS