I don't want the NSA to perform domestic civilian law enforcement functions any more than I want the army to, for similar reasons.
The type of oversight applied to military and foreign intelligence action is fundamentally political. Whether and how to impose sanctions against, spy on, or invade another country are questions for congress and the president in the US and rarely involve the courts.
The type of oversight applied to law enforcement is fundamentally legal. Courts apply existing law to the set of facts in a given case. Congress and the president do not get involved directly and any attempt by political officials to influence the outcome is scandalous, if not criminal.
So should the NSA hack Iran's immigration database to see who's been going there? Perhaps. Should anything discovered that way be admissible in a criminal trial? Almost certainly not. Should the NSA tip off the FBI about domestic crimes? Mostly no, I think, but perhaps in a very limited fashion when the crimes involve national security or foreign policy. Gathering evidence that can be used in a criminal trial would still be the responsibility of the FBI.