My mouth has started slanting to the right?

Possible stroke: This could be a symptom of stroke. You have to go to the er immediately to get evaluated.

Facial nerve change: Provided you didn't get new dentures or your teeth haven't been altered you likely have a palsy of the facial nerve. It is commonly called bells palsy and in general reverses in time unless due to a severe trauma to the nerve. Often caused by a viral illness. I recommend a neurological consult. Since its limited to your mouth and not the entire half of the face and eyelid i don't suspect stroke.

TMJ DiscDisplacement: Opening and closing movements should be accomplished in a straight line without deviation or deflection of the mandibular midline. Disc displacement conditions are characterized clinically by a limited range of motion on opening and a marked deflection to the affected side. Rec. See an orofacial pain specialist for evaluation and treatment if needed.