> why we need an integrator in DC servo instead of just an error correction amplifier?
As far as the "error correction" is concerned, our audio is an "error". It would try to reduce this error to zero. No audio output.
What we really want is: DC is an error, audio is not an error. The simplest way to define what is DC and what is audio is with a low-pass filter. And the most extreme low-pass is the integrator, because it tends to infinitely high gain (error reduction) at infinitely low frequency (DC).
> How to implement DC servo/offset compensation in a fully balance system?
You have errors in two terms, you need two servos. There may be some very clever way to share parts between two servos, but it still has at least two inputs, two outputs, and two gain functions. If you are not going into mass production, two or even three servos is the most direct and maybe-successful way to approach it.