Only Rome has the authority to approve an annulment. Once given, it must be accepted.
No, this is not correct. A decree of nullity is declared by a diocesan tribunal (court of the first instance) following investigation into a matrimonial case. Formerly, a case was sent to a court of second instance, usually the tribunal of another diocese within the same ecclesiastical province, often the metropolitan see itself. However, this second step of a secobd court was eliminated in the 1984 Code of Canon Law by procedural changes enacted by Bergoglio a few years ago. But, no, "Rome" does not alone have the authority to "approve" an annulment. This is a power of an ordinary (diocesan bishop or equal to diocesan bishop in jurisdiction) that is exercised through a canonically-constituted tribunal. The Roman Rota is a court of appeals for matrimonial cases for decisions rendered at the local level. It is not a court of first instance. Nor does the Holy Father serve as a judge of first instance, but only as an appellate judge (at least in canonical practice).