Do my 14 and 16 year olds need any other form of identification besides a birth certificate?

Nope, that's all they need. Just make sure it's a government issued one...issued by the city, county or state...whichever one does it in their state of birth. No hospital ones, no "notification of birth registration".

We are travellin to Aruba with our two sons in January. The are both adopted and all we have for one is a notification of birth because the new birth certificate has not been issued yet. We do have court papers stating that we are his legal parents.

Will this work? Any suggestion.

A notification of birth will not be sufficient. It must be a city, county or state issued birth certificate with the raised seal or watermark. If it doesn't arrive before your departure, take the court papers with you and you should be fine. Perhaps the attorney who handled the adoption could make a call and speed up the process for you of receiving the birth certificate.
Or call the department that issues them in the state of adoption and see if the adoption is on file yet so they can send you the official certificate. It's possible they have the documents to issue it and are just "sitting" on them but will move on it when you explain why you need it soon. If you wind up having to take the court documents just be sure to lock them up in your room safe with the rest of your families travel documents.

One thing that really helped us in lieu of passports (which we will have next time) when traveling with three children were state issued photo ID cards for each child. When the agent looked at the raised seal birth certificate, he/she could also see a photo. Dept. of Motor Vehicles in your state can issue these if you don't have time now to get passports. . good luck and have a blast with your kids!!!