In a circuit, electric field exerts force on electrons, so they must accelerate. Every text book I have read, points that electrons move with a constant drift velocity. How can this happen? Does Newton's law not apply there?
Yes Newton's law is applied here, and the electrons accelerate in response to the electric field. However, the electrons also undergo collisions with the atoms of the conductor, so on an average, they acquire an initial velocity of zero just after each collision. The electrons then acquire a final velocity $\vec v_f$ before the next collision and the average value of this final velocity is termed as the drift velocity, which is found to be a constant value, simply because the average time interval τ between each collision (called the relaxation time) is very small
Newton's law does apply but the electrons in the circuit aren't moving in a straight line like a car on a straight road for example. The motion of the electron is rather chaotic, they bounce around in the crystal and their average velocity is considered to be proportional to the Electric field.