Suppose we have a SDE in the form of $$dX_t = f(t, X_t, W_t)dt + g(t)dW_t$$ In my case, I am able to find one solution for this equation. Is there any theorem which can prove that this solution is unique?
More specifically, my SDE is $$dX_t = (\mu - \sigma \beta Z_t)dt + \sigma \sqrt{1 + 2\beta t}dW_t $$ where $Z_t = e^{-\beta t}\int\limits_0^t{e^{\beta u}\sqrt{1+2\beta u}dW(u)}$, and therefore depends only on $W_t$. The solution is $X_t = X_0 + (\mu - \frac{\sigma^2}{2}(1 + \beta t)))t + \sigma Z_t$
All the existence and uniqueness theorems I have found are about diffusion SDE, and therefore require $f(t, X_t)$ and $g(t, X_t)$. What can be done if $f$ contains $W_t$ component?
I'm not sure, but what if you redefine $X^1_t$ as your original SDE and $X^2_t$ as $W_t$.
Then rewrite it as a 2D diffusion SDE, with $X_t = (X^1_t,X^2_t)$. Let $$ \mu = \begin{bmatrix} f(X_t,t)\\ 0 \end{bmatrix} \;\;\;\;\&\;\;\;\; \sigma = \begin{bmatrix} g(X_t,t)\\ 1 \end{bmatrix} $$ So now you have a 2D SDE: $$ dX_t = \mu(X_t,t)\,dt + \sigma(X_t,t)\,dW_t $$ which is a standard Ito process.
Now you can maybe apply the multi-dimensional versions of the existence and uniqueness lemmas (e.g. this).