No shaking is needed. The sodium chlorite goes into solution immediately and does not separate out. You can give it a shake when you first add the drops, but there is no worry after that.
Sodium chlorite is a salt. To demonstrate this to yourself you can take a glass of water and take a salt shaker and shake a little salt into the glass. After the salt dissolves, it remains in solution and does not stratify. Now if you at a whole bunch this changes, but you are only adding a very small amount.