In Section 7.10.2 the free-free timescale quoted in the first paragraph is the correct one and the revision is wrong; random walking makes no change to the answer when the correct Sqrt(N) factor is included. Therefore absorption ceases to be important near the peak of the CMB spectrum when the temperature falls below 104 eV, a few days after the Big Bang (it remains important for much longer in the long-wavelength tail of the spectrum).
In Section 7.10.3 the final answer is correct because the premise is wrong. The expression quoted in the second paragraph is the classic Compton scattering formula, applicable to stationary electrons. In the early Universe the electrons were rapidly moving, and the Compton shift is swamped by the effects of the Doppler shift to and from the rest frame of the electrons, which gives a fractional frequency shift df/f of order v/c. A little algebra shows that this is of order the square root of the expression quoted by Raine & Thomas. This cancels out the mistake about random walks to give the right answer.