The classical electromagnetic waves as modeled by Maxwell's equations solutions, are one framework, the classical one.
Electrons and photons are elementary particles and are in the quantum mechanical framework.
The classical framework, emerges smoothly from the quantum framework at the overlap kinematic regions. One cannot use the same mathematics without critical thinking.
The electron does not absorb a photon. A free electron interacts with a photon according to quantum electrodynamic rules, and it is not a classical electrodynamics acceleration or deceleration. An electron bound to matter changes energy levels and the photon is absorbed or emitted from the atom/molecule/lattice. All these interactions are calculable with Feynman diagrams.
For example these are the first order diagrams that contribute to the free electrom photon interaction that can be used to give the probability of interaction and statistical plots of energy and momentum.
How classical fields emerge from quantum is a story that needs the mathematics of quantum field theory.
The classical observation that accelerating charges create electromagnetic waves as well as changing magnetic fields may also do, is beautifully explained with the classical electrodynamics of Maxwell's equations.