Just want to make sure you are using the ArcGIS Runtime SDK for Java as in this:
https://developers.arcgis.com/java/
The reason I ask is that there isn't a getPolygon method on the JavaFX control in this API.
Also want to double check what you mean by "drag and drop"? Are you just referring to panning the map to another location?
If you are after the current extent of you map, then you will be interested in the map's viewpoint. The code below shows how you could use it.
mapView.addDrawStatusChangedListener(listener -> {
System.out.println("draw status" + listener.getDrawStatus());
Envelope extent = (Envelope) mapView.getCurrentViewpoint(Viewpoint.Type.BOUNDING_GEOMETRY).getTargetGeometry();
if (listener.getDrawStatus() == DrawStatus.COMPLETED) {
System.out.println("extent " + extent.toString());
}
});
Does this help?