Which singular perturbation method should be used for this system? Consider the system
$$ \varepsilon \dfrac{dx}{dt} = -(x^3 - ax + b)$$
$$ \dfrac{db}{dt} = x - x_a$$
where $\varepsilon \ll 1$.  Applying regular perturbation methods isn't suitable because when $\varepsilon = 0$, the system turns into a differential - algebraic system.
Which singular perturbation method is best for problems similar to this?

I would recommend to use geometric singular perturbation theory, see
C. Kuehn, Multiple Time Scale Dynamics, Springer, 2015.
In this case, the so-called reduced slow system yields the (invariant) critical manifold
\begin{equation}
 C_0 = \left\{ (x,b) \vert b = x(a-x^2) \right\},
\end{equation}
with the slow flow on that manifold determined by $\frac{\text{d} b}{\text{d} t} = x-x_a$.
The reduced fast system, in the fast time variable $\tau = \frac{t}{\epsilon}$, we see that $b$ is (to leading order) constant in this time variable, while $x$ obeys
\begin{equation}
 \frac{\text{d} x}{\text{d} \tau} = a x - x^3 - b.
\end{equation}
Addition: For this particular model, it is very instructive to look into (singular perturbation) analysis on the Fitzhugh Nagumo model, or the van der Pol equation. Kuehn's book is a good starting point for that, too.
