Keep in mind, balloting the US is based on place of residence. You’ll have a different set of elections to vote in from municipality to municipality (or ward to ward).
So you need to have a specific address in the US.
This makes sense. You shouldn’t be able to vote for an alder person in a different ward. But it requires a specific, “in US” residence.
If you don’t have a specific address in the US, (you’re permanently residing outside the country) your last state of residence should be able to provide you with “President Only” ballots.
I’ve never heard of this actually happening though.
The US embassy has no involvement in any of this whatsoever. They will refer you to the clerk of courts (or whatever).
Realistically, do everything you can in the US (including absentee voting if possible).
Edit: keep in mind, many places have over a dozen distinct election events over a 4-year cycle. Presidential, presidential primary, mid-term, mid-term primary, non-partisan, non partisan primary, local school board, etc.
Some states combine this into as few elections as possible, some as many elections as possible.
I once tried to vote in every election. Lack of publicity made this impossible. For many minor elections, there was no public info on the candidates for, say, clerk of court primary elections.