This problem seems to always occur when installing aftermarket railings, balusters, and newels. Seems that installing them with the construction of the stairs are much stronger.
My speculation and experience tells me that this will only get worse and the rails and balusters will not lend any support to the newel posts. The lean or movement will get progressively worse as time and force is applied.
This is about like when I am out building a deck. I always ask if an arbor is intended to be built. If no, I let them know that if any idea of an arbor is desired, its better to build it before the deck so that the posts can be in the earth, cemented and an intrigral part of the deck.
An arbor build after the deck will not be as secure as the one with the deck.
Same scenerio with railings.
You might strengthen the newels if they were bolted to the first riser of the first step and a cut out is made to allow it to fit there and bolt it to that riser. The riser also needs to be reinforced to accept it. The top newel is another animal. you migh want to attach it to the very top riser and knotch the tread for its fit.
A newel must be steady, tight and very secure. its a stand alone component of the railings. Sure the railing will secure it from sideways movement but the opposite side ways movement cannot be controlled unless its fitted properly.
I would give this alot of thinking on your newels. You do not want any movement from them.