Robert Bryant gives an excellent answer. Let me add some more comments.
Your question is solved by Douglas who was awarded the Fields medal due to his work on Plateau's problem. His paper, however, is a bit hard to read since it is written nearly 90 year ago. Here's a modern solution due to Jost. Douglas's result on your question can be roughly described as the following: $\DeclareMathOperator{\area}{Area}$
Let $a_i=\inf\{\area(\Sigma_i)\mid \Sigma_i \text{ is a disk spanned by } \gamma_i\}$. Suppose that there's an annulus $\Sigma$ spanned by $\gamma_1 \cup \gamma_2$ such that $$\area(\Sigma) < a_1+a_2,$$ then there's a minimal annulus spanned by $\gamma_1 \cup \gamma_2$. The criterion above is called the Douglas criterion, and the strict inequality is essential.
A final remark. To solve this question, you can't fix an annulus $A=\{1 < |z| < 2\}$ since conformal annuli are not conformally equivalent to each other. The key ingredients are the following.
Let $\Gamma= \gamma_1 \cup \gamma_2$. Let \begin{equation} \begin{split} \mathcal{F}_\Gamma = \{ & u\mid u \in C(\bar{\Sigma}) \cap W^{1,2}(\Sigma, \mathbb{R}^3), \Sigma=\mathbb{S}^1 \times (0,s), \\ & u|_{\partial \Sigma}: \partial \Sigma \to \Gamma \text{ is a homeomorphism. }\}. \end{split} \end{equation}
- Take a sequence of functions $u_k \in \mathcal{F}_\Gamma, u_k: \mathbb{S}^1 \times (0, s_k) \to \mathbb{R}^3$ such that $E(u_k) \to \inf$, where $E(u_k)$ is the energy of $u_k$.
- Show that $\{s_k\}$ has positive lower bound. This is proved simply by Cauchy's inequality.
- Show that $\{s_k\}$ has finite upper bound. This is guranted by Douglas's criterion, which is the key idea of Douglas. Then we get a convergent subsequence $\Sigma_k \to \Sigma = \mathbb{S}^1 \times (0, s)$.
- Show that $u_k|_{\partial \Sigma}$ is equicontinuous. This is obtained by Douglas's criterion and Courant-Lebesgue's lemma. Therefore, we can suppose that $u_k|_{\partial \Sigma}$ converges uniformly.
- Replace each $u_k$ by the unique harmonic function with the same boundary value, which is denoted by $v_k$. Then $v_k$ is also an energy minimizing sequence and $v_k$ converges uniformly. The limit function is denoted by $v$.
- Show that $v$ attains the infimum of energy, and then attains the infimum of area.