Question

do electrons really radiate energy?

Answer 1

Electrons emit something called Bremsstrahlung radiation due to the non-zero "response delay" of the nearby electromagnetic field upon acceleration. This is the basis of virtually all electromagnetic radiation that exists.

https://en.wikipedia.org/wiki/Bremsstrahlung

Answer 2

This is the best example I can show you:
http://www.qsl.net/g3yrc/radiate/Image32.gif

As those "kinks" in the field lines propagate, you have an electromagnetic wave, carrying with it some of the energy from the original electron.

Answer 3

Electron emit energy when it returns from higher electronic state
To ground state in atom. It gives off extra energy as radiation.
It is not radiating all the time when circulating nucleus in ground-state level.