Based on the comments, I think what you mean by the global Waldspurger packet of $\pi$ is the Vogan packet, i.e., the set of all cuspidal representations $\pi_D$ of a quaternion algebra $D/F$ such that $\pi_D$ corresponds to $\pi$ in the sense of Jacquet-Langlands. This is indeed finite as you say in your comment.

However this finiteness is unrelated to the question of how many twists of $\pi$ will have nonvanishing central $L$-value. Let $\chi_E$ be the quadratic character associated to a quadratic extension $E/F$. Waldspurger relates $L(1/2, \pi)L(1/2, \pi \otimes \chi_E)$ to a period $P_E$ of some $\pi_D$ in the Vogan packet. As $E$ varies, the period changes, as does the "right" choice for the $D$ on which to choose the $\pi_D$. In particular, infinitely many periods (and thus $L$-values) may vanish and infinitely may be nonzero on the same $\pi_D$.

So there is no contradiction between finiteness of the Vogan packet and the nonvanishing of infinitely many twists.

Revised. The global Waldspurger packet of $\pi$ is indeed finite, as you say in your comment. It's elements are metaplectic representations which are in bijection with the Vogan packet of $\pi$, i.e., me the set of all cuspidal representations $\pi_D$ of a quaternion algebra $D/F$ such that $\pi_D$ corresponds to $\pi$ in the sense of Jacquet-Langlands. There are finitely many relevant $D$'s, which are determined by the finite set of places at which $\pi$ is discrete series.

However this finiteness is unrelated to the question of how many twists of $\pi$ will have nonvanishing central $L$-value. Let $\chi_E$ be the quadratic character associated to a quadratic extension $E/F$. Waldspurger relates $L(1/2, \pi)L(1/2, \pi \otimes \chi_E)$ to a period $P_E$ of some $\pi_D$ in the Vogan packet. As $E$ varies, the period changes, as does the "right" choice for the $D$ on which to choose the $\pi_D$. In particular, infinitely many periods (and thus $L$-values) may vanish and infinitely may be nonzero on the same $\pi_D$.

So there is no contradiction between finiteness of the Vogan packet and the nonvanishing of infinitely many twists. I think the confusion is arising from an assumption that the theta lifts $\Theta(\pi \otimes \chi_a)$ give different automorphic representations for different $a \in F^\times/F^{(2)}$, which is not true.