There in fact typically is an equilibrium of diffusion for both oxygen and for carbon dioxide by the time the blood finishes gas exchange in the alveoli of the lung. One confusion in the OP is that there is no real meaning for a chemical equilibrium between carbon dioxide and oxygen, as they are separate chemical entities. Carbon dioxide in the lung capillary blood equilibrates with carbon dioxide in the lung alveolar gas; oxygen similarly equilibrates between blood and gas. Each effectively on its own.
Yes, oxygen is continually being consumed and carbon dioxide being produced by the body, so from that perspective we have a whole body steady state. At overall equilibrium we are dead. But at the site of gas exchange in the lung, there is generally no difference in the chemical activity of either gas between the alveolar air and the capillary blood leaving the alveolus. So from a chemical perspective there is generally local equilibrium of both gases at that point.
In some circumstances this local equilibrium is not reached. The transport of carbon dioxide is very complicated, as it exists in several inter-convertible forms: dissolved carbon dioxide gas, a form called carbonic acid where it has chemically reacted with water, the bicarbonate and carbonate ions that come from dissociation of carbonic acid, and a carbamate form that is a reversible chemical bond with amino groups on proteins. The reaction with water is slow on its own; it is catalyzed by the enzyme carbonic anhydrase in red cells. If that enzyme is inhibited (for example, by the drug acetazolamide) then there might not be chemical equilibrium of total carbon dioxide between alveolar air and capillary blood as there might not be enough time, as blood flows through the alveolus, for all the other forms to convert to the dissolved gas that can diffuse from blood into alveolar air.
Another complication is that transport of oxygen and carbon dioxide in the blood are somewhat interdependent. Acidification by carbon dioxide (lower pH is more acid) affects the ability of hemoglobin to bind oxygen, tending to release more oxygen. Nevertheless, in most circumstances in humans, both gases are individually at equilibrium between alveolar air and the blood that leaves the alveolus.
Finally, the continued presence of carbon dioxide in the body is a normal, healthy part of physiologic functioning. It does have to be kept in balance, but it does have to be there.