Should I be able to shift my septum side to side?

The septal cartilage sits on top of bone, the maxillary crest. Sometimes it can get displaced from trauma. It makes sense that you can move it. It is not a surgical necessity to fix it, however if there are functional breathing issues than fixing it makes sense.

There can be slight movement, but if you are able to significantly shift the septum, or there is pain associated with the movement, I recommend you see a specialist. Particularly in light of your history and symptoms, it is a good idea to have a doctor examine you. I encourage you to schedule a consultation with an experienced, board-certified facial plastic surgeon who has been through ENT training as well as plastics. This specialist can assess the reason for the movement and create a treatment plan should you decide to make corrections to the nose.

Sometimes with or without trauma the cartilage of the septum or nose may change and become more 'moveable' in time.. occasionally the mobility of the septum can even cause a clicking noise that can be caused by moving the cartilage.. if the area is of concern I would ask your rhinoplasty surgeon what can be possibly done about the area.. 

The upper portion of the structure of the nose is bone, and the lower portion is cartilage. Rhinoplasty can change bone, cartilage, skin or all three. Talk with your surgeon about whether rhinoplasty is appropriate for you and what it can achieve.

While the septum can be moved around a bit just by pushing, I am concerned when you discuss the pain associated with the movement. Due to your history of nose injury and your symptoms, I recommend you see a specialist to determine if correction would be of benefit to you, as well as timing of any intervention.