We study robust parameter-efficient fine-tuning (PEFT) techniques designed to improve accuracy and generalization while operating within strict computational and memory hardware constraints, specifically focusing on large-language models (LLMs). Existing PEFT methods often lack robustness and fail to generalize effectively across diverse tasks, leading to suboptimal performance in real-world scenarios. To address this, we present a new highly computationally efficient framework called AdaZo-SAM, combining Adam and Sharpness-Aware Minimization (SAM) while requiring only a single-gradient computation in every iteration. This is achieved using a stochastic zeroth-order estimation to find SAM's ascent perturbation. We provide a convergence guarantee for AdaZo-SAM and show that it improves the generalization ability of state-of-the-art PEFT methods. Additionally, we design a low-rank gradient optimization method named LORENZA, which is a memory-efficient version of AdaZo-SAM. LORENZA utilizes a randomized SVD scheme to efficiently compute the subspace projection matrix and apply optimization steps onto the selected subspace. This technique enables full-parameter fine-tuning with adaptive low-rank gradient updates, achieving the same reduced memory consumption as gradient-low-rank-projection methods. We provide a convergence analysis of LORENZA and demonstrate its merits for pre-training and fine-tuning LLMs.