Originally Posted by Often1

You can depart the US on any passport which the receiving country recognizes. The US simply requires a passport, but there is no outbound passport control. Your air carrier is looking for a document which will get you admitted at the destination. Thus, presuming you are headed to a country which recognizes Mexico and does not also require a visa of Mexican nationals, you are all set.

On return, it is a bit dicier. While US law requires that USN's travel to the US on a US passport, that provision is not enforced. Additionally, as a USN, you cannot be denied entry to the US. However, this does not mean that it will be a pleasant experience at CBP and your air carrier will be looking for the documents which support a US entry for a Mexican national, e.g., a visa. Thus, you may be denied boarding.

However, this gives rise to the question of why you can "no longer" obtain a US passport. Without knowing the reason for that statement by you, it is hard to really drill down. Can you provide some information?