Let's begin by pointing out the following: you will not find monotone examples for the simple reason that a nontrivial such deformations creates a class of nonzero symplectic area, while the Chern class is always vanishing. The best you could hope for is Calabi-Yau, and such examples indeed exist. However, I know of no examples where a closed Lagrangian has a nonvanishing Floer homology.

Now a general observation: In the case when $T^*N \setminus N$ has no second cohomology with $\mathbb{R}$-coefficients, e.g. if $N=S^2,$ then for small closed forms $\sigma$ one can use Moser's trick to show that any compact Lagrangian submanifold of $T^*N \setminus N$ is preserved (up to smooth isotopy) after turning on a sufficiently small magnetic potential.

A more concrete example: taking $\sigma$ to be the area form on $S^2,$ we obtain the total space of the line bundle $\mathcal{O}(-2)$ on $\mathbb{C}P^1$ with its standard Kähler form. (The first reference coming to my mind is 2.4A in [Y. Eliashberg and L. Polterovich; Unknottedness of Lagrangian surfaces in symplectic 4-manifolds] but maybe there is something more to the point). Unlike $T^*S^2$, the latter symplectic manifold is an open toric Calabi-Yau manifold. Unfortunately, according to Theorem 5 in [Ritter; Floer theory for negative line bundles via Gromov-Witten invariants], its symplectic homology vanishes: this twisted cotangent bundle therefore contains no Lagrangians with interesting Floer homology. See [Ritter-Smith; The monotone wrapped Fukaya category and the open-closed string map] where a closed-open map is constructed in this setting.

A side note: if you compactify a subset of the total space of $\mathcal{O}(-2)$ to the Hirzebruch surface $F_2(\alpha)$ as studied in [Fukaya-Ohta-Ono-Oh; Toric degeneration and non-displaceable Lagrangian tori in $S^2\times S^2$] then Fukaya category actually becomes nontrivial.