Here is how to register to vote:
https://www.fvap.gov/citizen-voterAFAIK you always use the last US address you lived at and you can't change that. This becomes your "voting address". Even if strangers are now living there.
I have to renew this registration every year or two.
I think the Democrats Abroad is more like a club/pressure group.
Note that some states will consider you liable for their income tax (a resident for tax purposes) if you vote for non-Federal offices. I've never gotten to the bottom of it for Arizona, it's a murky area it seems. To be safe I requested a federal-only ballot (my request was ignored...) and so I ignore all the non-Federal offices.
If you need help with registering to vote using the FVAP site post back here. It's pretty easy to do.