Dependency pairs (DPs) are one of the most powerful techniques for automated termination analysis of term rewrite systems. Recently, we adapted the DP framework to the probabilistic setting to prove almost-sure termination ( \(\texttt{AST}\) ) via annotated DPs (ADPs). However, this adaption only handled \(\texttt{AST}\) w.r.t. the innermost evaluation strategy. In this paper, we improve the ADP framework to prove \(\texttt{AST}\) for full rewriting. Moreover, we refine the framework for rewrite sequences that start with basic terms containing a single defined function symbol. We implemented and evaluated the new framework in our tool AProVE.