The authors introduce a new variant of domination in graphs called weak double Roman domination (WDRD), which generalizes the well-studied concept of double Roman domination (DRD) by relaxing certain constraints. Given a graph \( G = (V, E) \), a WDRD-function is a labeling \( f: V \to \{0,1,2,3\} \) that satisfies the following condition: every vertex \( v \) with \( f(v) \leq 1 \) must have a moving neighbor \( u \), meaning a vertex with \( f(u) \geq 2 \), such that if we transfer one unit from \( u \) to \( v \), the new function \( g \), defined as \(g(v) = f(v) + 1, \quad g(u) = f(u) - 1, \quad g(x) = f(x) \text{ for all } x \in V \setminus \{u, v\}\), ensures that no vertex is doubly unprotected. A vertex \( v \) is called doubly unprotected with respect to \( f \) if the total sum of function values in its closed neighborhood is at most 1, i.e., \( f(N[v]) = \sum_{x \in N[v]} f(x) \leq 1.\) This means that \( v \) itself and all its neighbors together have at most one unit of defense, making it particularly vulnerable. The weak double Roman domination number \( \gamma_{\mathrm{wdR}}(G) \) is the minimum possible weight of a WDRD-function, where the weight of \( f \) is given by \( \sum_{v \in V} f(v).\) The study establishes upper and lower bounds for the weak double Roman domination number \( \gamma_{\mathrm{wdR}}(G) \), showing its relation to other domination parameters, such as double Roman domination and weak Roman domination. The paper also provides exact values for paths, cycles, and ladders and proves that the decision problem associated with WDRD is NP-complete, even for bipartite and chordal graphs.