Spoofing-robust automatic speaker verification (SASV) seeks to build automatic speaker verification systems that are robust against both zero-effort impostor attacks and sophisticated spoofing techniques such as voice conversion (VC) and text-to-speech (TTS). In this work, we propose a novel SASV architecture that introduces score-aware gated attention (SAGA), SASV-SAGA, enabling dynamic modulation of speaker embeddings based on countermeasure (CM) scores. By integrating speaker embeddings and CM scores from pre-trained ECAPA-TDNN and AASIST models respectively, we explore several integration strategies including early, late, and full integration. We further introduce alternating training for multi-module (ATMM) and a refined variant, evading alternating training (EAT). Experimental results on the ASVspoof 2019 Logical Access (LA) and Spoofceleb datasets demonstrate significant improvements over baselines, achieving a spoofing aware speaker verification equal error rate (SASV-EER) of 1.22% and minimum normalized agnostic detection cost function (min a-DCF) of 0.0304 on the ASVspoof 2019 evaluation set. These results confirm the effectiveness of score-aware attention mechanisms and alternating training strategies in enhancing the robustness of SASV systems.