Given how often I've had the government send things to the wrong address for me, I am fairly sure that they don't know where I truly live (most of the time), even when I gave them the address.

Unlikely. I know of a lot of people who live in New Jersey but register their auto in another state under a relative's address for cheaper insurance. I also know people who use an old address or the address of an acquaintance/relative so their child can attend a different school district.

That just implies that communication in government is poor, particularly if it is a local or state government that wants to know something from the federal government. I am sure the IRS would have no problem finding those people if they were so inclined.

I became a South Dakota resident in January (while I was in CA.). Despite only having spent a week in the state last summer. I haven't been back since then.