How can I get US Congressional District code by latitude and longitude OR user address? I have already downloaded Congressional district shapefile from Census files.
Create points from your lat/lon values - see e.g. here how to do it.
On the point layer, apply this expression:
overlay_within('district', "code")and replace
districtwith the name of the polygon layer containing the districts and
codewith the attribute field containing the US Congressional District code.