I think the problem you are having is thinking that Newton's third law means that the equal and opposite forces cancel each other in terms of the effect one has on the other. Sometimes that's true, sometimes not.
Let's say you push against a fixed wall and neither of you move. The wall does not move because the ground exerts a force on the wall equal and opposite to the force you exert on the wall. Therefore there is no net force on the wall. You don't move because the ground exerts a force (friction) on your feet equal and opposite to the force the wall exerts on you. So there is no net force on you.
If you are driving a car at constant velocity and an insect hits your windshield, the insect exerts an force on your windshield equal and opposite to the force your windshield exerts on it. The effect on each is dramatically different. The deceleration of the car due to the force applied by the insect is minuscule compared to the acceleration of the insect due to the same force applied by the car. This is because you need to couple Newton's third law with Newton's second law, $F=ma$.
The force the insect exerts on the windshield is
The force the windshield exerts on the insect is
Where $M$ is the mass of the car, $m$ is the mass of the insect, $a$ is the acceleration of the car, $A$ is the acceleration of the insect. Comparison of the magnitudes is
So while the sum of the forces equals zero, there is a net force on each due to the other causing a different acceleration of each.
Hope this helps.