Isometric rigidity and Fraïssé properties of Orlicz sequence spaces

We provide an approximate version of a rigidity result by Randrianantoanina: for a large class of Orlicz sequence spaces, almost isometric embeddings almost preserve disjointness. In specific cases, we can even prove that such embeddings almost preserve basic vectors. As a consequence, we prove that some Orlicz sequences spaces are guarded Fraïssé but not $\omega$-categorical; moreover, they do not contain copies of ${\mathrm{\ell }}_2$ and their age is not closed. This answers a question of Cúth-de Rancourt-Doucha.