If I understand correctly, you can do this by using a flag. But you want to set the flag so exactly one row for each existing address has the flag set.
ALTER TABLE t ADD address_for_duplicates boolean DEFAULT true NOT NULL;
Then, for existing rows, I will assume that you have a primary key, pk:
update t
set address_for_duplicates = (seqnum = 1)
from (select t.*, row_number() over (partition by address order by pk) as seqnum
from t
) tt
where tt.pk = t.pk;
Now add a filtered unique index:
create unique index unq_t_address_some_duplicates
on t(address)
where address_for_duplicates;
This will prevent existing addresses from being duplicated (again) as well as new addresses.