The Strong $(2,2)$-Conjecture asks whether, for all connected graphs different from $K_2$ and $K_3$, we can assign red and blue labels with value $1$ or $2$ so that no two adjacent vertices have the same sum of incident red labels or the same sum of incident blue labels. This conjecture, which can be perceived as a generalisation of the so-called 1-2-3 Conjecture, as, thus far, been proved only for a handful number of graph classes. In this work, we prove the Strong $(2,2)$-Conjecture holds for more classes of graphs. In particular, we prove the conjecture for cacti, subcubic outerplanar graphs, graphs with maximum average degree less than $\frac{9}{4}$, and Halin graphs, among others.