If I was involved in a front end crash and tore my rotator cuff when my airbags did not deploy and my insurer found out that there was a recall issued on my airbags/airbag system, do I have a case?

You have a product liability claim against the auto manufacturer and airbag manufacturer.  Prior to filing a lawsuit, it may be possible to settle the case with the insurance carriers for both the auto manufacturer and the airbag manufacturer.
Notify both insurance carriers in writing that you will be filing a personal injury claim.
When you complete your medical treatment and are released by the doctor or are declared by the doctor to be permanent and stationary, which means having reached a point in your medical treatment where no further improvement is anticipated, obtain your medical bills, medical reports, and documentation of wage loss.  Your personal injury claim filed with the insurance carriers for the auto manufacturer and airbag manufacturer should include those items.
Compensation for the medical bills is straight reimbursement.  The medical reports will document the nature and extent of your injury and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  Compensation for wage loss is straight reimbursement.
If the case is settled with the insurance carriers for the auto manufacturer and airbag manufacturer, NO lawsuit is filed.
If you are dissatisfied with settlement offers from the insurance carriers, reject the settlement offers and file a lawsuit against the auto manufacturer and airbag manufacturer for negligence and strict liability.
Negligence is the failure to exercise due care to produce a product that is not defective.  Strict liability is liability whether or not due care was exercised.  Negligence and strict liability are separate causes of action claims against both the auto manufacturer and airbag manufacturer.
If the case has been settled with the insurance carrier for one of the parties auto manufacturer or airbag manufacturer, only name the party with whom the case has NOT settled as a defendant in your lawsuit..  If the case has NOT settled with either party, name both as defendants.  If the case has settled with the insurance carriers for both parties, NO lawsuit is filed as mentioned above.
If the case is NOT settled, your lawsuit for negligence and strict liability must be filed prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.