3 manifolds – Parametric general position theorem for foliations

The situation is the following: let $M$ be a manifold endowed with a smooth foliation $mathcal{F}$ of codimension one (suppose orientable, transversely orientable) and let $F_t : S rightarrow M$ be a one parameter family of smooth inmersions. Is it true that there exists an arbitrarily small homotopy from $F_t$ to a family of inmersions $G_t$ such that $G_t$ is in general position with respect to $mathcal{F}$? Furthermore can I say something about the number of singularities at each step and how are the transitions? (maybe this is a lot to ask). More concretely I’m thinking in $M = mathbb{T}^3$ and the embedded surfaces being $mathbb{T}^2 times {pt}$.

Pd: by general position I mean that it is transversal except at finitely many points where the induced foliation on $S$ has Morse type singularities.