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Deep learning-based methods deliver state-of-the-art performance for solving inverse problems that arise in computational imaging. These methods can be broadly divided into two groups: (1) learn a network to map measurements to the signal estimate, which is known to be fragile; (2) learn a prior for the signal to use in an optimization-based recovery. Despite the impressive results from the latter approach, many of these methods also lack robustness to shifts in data distribution, measurements, and noise levels. Such domain shifts result in a performance gap and in some cases introduce undesired artifacts in the estimated signal. In this paper, we explore the qualitative and quantitative effects of various domain shifts and propose a flexible and parameter efficient framework that adapts pretrained networks to such shifts. We demonstrate the effectiveness of our method for a number of reconstruction tasks that involve natural image, MRI, and CT imaging domains under distribution, measurement model, and noise level shifts. Our experiments demonstrate that our method achieves competitive performance compared to independently fully trained networks, while requiring significantly fewer additional parameters, and outperforms several domain adaptation techniques.