Eduardo,
What you are wanting to do is possible but pretty complicated.
Here is my recommended workflow steps that you would have to do:
1. Use the edit toolbar inorder to add your new feature.
Edit without editor widget | ArcGIS API for JavaScript
2. In the draw-end function before the selectedTemplate.featureLayer.applyEdits you would need to query the districts layer to see if the point you placed intersects.
3. For that you would use a QueryTask and a Query class that would use the geometry from step two and the SPATIAL_REL_INTERSECTS
4. Once you have results from that query you can get the results[0].attributes.District and put that in a var.
5. Then update the newAttributes array to replace the existing District value with the new on retrieved.
6. Then you can finally the selectedTemplate.featureLayer.applyEdits.
If you can not comprehend this workflow then this is probably something you will need to hire a developer/consultant company for as I mentioned this is possible but is not something that is easy.