I’m sure it depends on the different state’s laws; but here in NJ, you are offered to register to vote at Motor Vehicles when you get your Driver’s License.

The notice you got in TN was probably just an outreach they do in advance of Election Day to “get out the vote”. I’ve never heard of anyone having to notify the state, county or municipality that they are leaving when they move away – just the USPS change of address – so I think it doesn’t really matter.

When I moved to TN I was called for jury duty in the state I had just left. I already had been living in TN for a few months. I had my license there, was registered to vote, and owned property.

The states don’t tell each other I guess.

When I registered in Oregon they asked me what state and county I was previously registered in. I assumed that meant they were going to notify them. Here, the rule at DMV is you are registered unless you opt out and all ballots are vote by mail.