Each State handles its own qualified voter list and is responsible for maintaining its accuracy. You are correct that there is no Federal electoral roll; each State must administer its own elections and that includes voter registration, although Federal law places certain restrictions and obligations on how the States conduct elections.

As far as place of residence: keep in mind that we do not only vote for President, but also for US Senators (for our State) and US Representative for our Congressional district (which is typically a small portion of one State) as well as State-level offices (Governor, State Senate & Reps). We also vote for county-level offices (top-level administrative, court, and law enforcement directors) and city or town-level offices (mayor, commissioner, police chief) and perhaps even board of directors of school districts, libraries, etc. Each State gets to decide the rules of what sort of municipalities can exist in its borders and how they function, and in turn, what sort of elected offices can exist within each municipality. To expect the Federal government to keep track of such nitty-gritty details is just too much micromanaging, so the Feds don't even bother.

If you're moving within a State, and you register with your new address, the State will (normally) recognize the old address as no longer valid and just change your voting precinct. Trying to vote in the old precinct will just get you turned away. (Or if you insist on voting there, you'll get a provisional ballot that won't be counted until and unless you can prove the voter roll was in error.)

If you're moving to a new State, you're usually asked to inform the new State that you were previously registered to vote in your old State, so the new State can inform the old State to remove you from the rolls. If you don't do that, however, you will remain on both rolls for some period of time.

Your old State may still delete you from the rolls after cross-referencing with change-of-address lists from the (State) registry of motor vehicles or the (Federal) US Post Office.

Some States exchange limited information on voter rolls, for the purpose of recognizing duplicates and removing old entries, and you may be removed from the old State that way as well.

Since duplicate voting is such an incredibly inefficient means to commit election fraud, and aggressive purging of voter rolls to "accidentally" disenfranchise certain groups is an extremely efficient means to commit election fraud, it's generally in the best interest of democracy to have a measured approach toward managing voter rolls.