Sodium carbonate does not react with oxygen or undergo an oxidation reaction. You can think about it this way: the sodium in sodium carbonate (Na₂CO₃) is already oxidized to its maximum oxidation state of Na+. Similarly, the carbon, as carbonate, is maximally oxidized (consider how carbonate is chemically very similar to/derived from CO₂, the maximally oxidized form of carbon combined with oxygen). So no further oxidation will occur.

Sodium bicarbonate does not react with oxygen. It reacts with hydrogen.

The way that it works is that the hydrogen in an acid (say, vinegar) replaces the sodium in the bicarbonate to form hydrogen carbonate, an acid consisting of two hydrogen atoms, one carbon, and three oxygens. Hydrogen carbonate can break up to become carbon dioxide and water. The carbon dioxide can then bubble off as a gas.

Baking soda sits quite nicely in the air without oxidizing. Baking soda has lots of oxygen already, because its chemical formula is NaHCO₃, which is a sodium ion [Na+] and a bicarbonate ion [HCO_3-] that has 1H + 1C + 3O atoms. Baking soda is sodium bicarbonate.

Because baking soda has so many oxygens, it can oxidize other molecules, but this is unusual: Although this is unusual, NaHCO₃ can act as an oxidizing agent. More information.