How to determine the ground state configuration for a Hamiltonian as a function of expansion in terms of some parameter in the Hamiltonian?

I have been learning about lattice gauge theories, in particular about the Ising gauge theory on the 2D square lattice. The Hamiltonian for a system with no matter fields is given by (for eg. from this book, Section 9.6, Field Theories of Condensed Matter Physics )
$$ mathcal{H} = – g sum_{vec{x},j} sigma^x_j(vec{x}) – frac{1}{g} sum_{boxed{}} prod_{vec{x},j in boxed{}} sigma^z_j(vec{x}) $$
where $sigma^x_j, sigma^z_j$ are the Pauli spin operators acting along a link labelled in the direction of the basis vector $vec{e_j}$ from the site $vec{x}$ and the $boxed{}$ refers to a plaquette of the lattice.

I am able to understand that when $g to 0$ or $infty$, what the ground state will be. For example, in the $g to infty$ limit, all the spins in the x-direction will align themselves in the ground state. However, I am not able to figure out how I should proceed, (i.e. to find the ground state configuration) when one of the terms in the Hamiltonian is not exactly zero, but something close to it (say, at $ O(g^2)$).