Mandibular Fracture: This is the most likely diagnosis given the context of a car accident, which can cause significant trauma to the face and jaw. A fracture would explain the shifting or instability of the lower jaw.

Temporomandibular Joint (TMJ) Dislocation: The force from a car accident could dislocate the TMJ, leading to jaw misalignment and shifting. This condition can cause significant pain and difficulty in moving the jaw.
Mandibular Dislocation: Similar to TMJ dislocation, the mandible itself could be dislocated from its normal position, causing the jaw to shift.
Soft Tissue Injury: Injuries to the muscles, ligaments, or other soft tissues around the jaw could also lead to instability or shifting, although this might be less common than bony injuries in the context of a car accident.

Cervical Spine Injury: Although the primary complaint is jaw shifting, it's crucial not to miss potential cervical spine injuries, which can be associated with facial trauma and have severe consequences if not promptly addressed.
Basilar Skull Fracture: This is a serious condition that can occur with significant facial trauma and might present with jaw instability among other symptoms. It requires immediate medical attention.

Pathological Fracture: If the patient has a pre-existing condition like osteoporosis or a bone tumor, a pathological fracture could occur with relatively minor trauma, leading to jaw shifting.
Osteonecrosis of the Jaw: Although rare and more commonly associated with long-term bisphosphonate use or radiation therapy, osteonecrosis could potentially lead to jaw instability in the setting of trauma. However, this would be an uncommon diagnosis in the acute setting of a car accident.