Your first stop needs to be at your vet's office. There are medications that can help with separation anxiety. I believe the most commonly used drugs for SA are Reconcile (fluoxetine hydrochloride) and Clomicalm (clomipramine hydrochloride). The drugs aren't a cure-all, but rather are meant to reduce the dog's anxiety while you work on re-training. It's next to impossible for a highly anxious, stressed dog to learn.
Be aware that even with medication to help and a diligent re-training program, it can take a very long time to help a dog with SA. It's a very difficult problem to deal with. It can be done, though.
The other thing that works for some dogs is to get another dog. Please note that I am NOT recommending you take that route. I advise anyone with a SA dog to get another dog only if they want another dog for him/herself. Because there's no guarantee adding another dog will help the dog with SA, and anyone who goes that route needs to be prepared to keep the second dog and give it an excellent home regardless.
An obedience class is always a good idea, and the confidence that comes with learning new commands can be somewhat helpful for SA dogs. But in general you shouldn't expect any huge improvement from a class. I would consider seeing a board certified canine behaviorist instead of looking for a trainer or class. A behaviorist is a licensed vet who's had special training in behavior issues. Your vet will hopefully be able to refer you to a behaviorist.