If you have two harmonic quantites $x,y$:
$$
x=\Re(Xe^{i\omega t})\\
y=\Re(Ye^{i\omega t})\\
$$
then the average over a period of the product is given by:
$$
\langle xy\rangle_T=\frac{1}{2}\Re (XY^*) =|X||Y|\cos\phi
$$
where $\phi$ is the phase difference between the signals. Interpreting the product as a form of power, the more in phase the signals are, the more power you get.
You can use this to do an energy balance of your system. The power of the force must equal the power dissipated in the stationary state since energy does not vary on average.
Thus the relative phase between velocity and force (or equivalently between position and force up to an additional phase) is important to calculate the injected power.
Mathematically, you get the equality of the powers:
$$
P=2\gamma\langle \dot x^2\rangle_T= \langle f \dot x\rangle_T
$$
from multiplying by $\dot x$ the equation of motion and taking the time average. Note that on the LHS, you get an additional term $\dot E$, the time derivative of energy, but its average value in the stationary state is zero.
Using the previous result, you get:
$$
2\gamma\omega^2 |X|=F\sin\sigma
$$
(there is a $\pi/2$ phase difference between $x$ and $\dot x$). Energy conservation thus gives a relation between amplitude $|X|$ and dephasing.
From this relation for example, you can see that the presence of damping forces dephasing in order keep a finite amplitude. In fact, it indicates that resonance, a maximum of $X$ is accompanied by a lagging of $\pi/2$.
Note that energy conservation alone does not give you both amplitude and dephasing, only a relation between the two. Therefore, you need some information beforehand about amplitude in order to use conservation of energy to deduce your dephasing and vice versa.
Hope this helps.
Answer to comment
You need to be careful, try to avoid energy conservation, but rather energy balance. Even if the averaged energy per period does not change, there is still dissipation and injection, i.e. fluxes. Therefore, there is global balance and not detailed balance (synonymous to energy conservation).
Within a period, there is not even energy balance. Recall that energy is:
$$
E = \frac{1}{2}\dot x^2+\frac{\omega_0^2}{2}x^2
$$
so using complex notation:
$$
x^2 = \frac{1}{2}\Re (X^2e^{i2\omega t})+\frac{1}{2}|X|^2
$$
you get:
$$
E = \frac{-\omega_d^2+\omega_0^2}{4}\Re(X^2e^{i2\omega_d t})+\frac{\omega_d^2+\omega_0^2}{4}|X|^2
$$
The second term is the constant value that you get from averaging. However, the first term shows that in general, $E$ fluctuates at frequency $2\omega_d$, ie twice as fast as a period.
Note that at resonance, $E$ is constant even within the period. However, the same caveat applies. There will still be dissipation and injection, so you only have global balance at each time. This is why you avoid saying that energy is conserved.
