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Solving the domain shift problem during inference is essential in medical imaging as most deep-learning based solutions suffer from it. In practice, domain shifts are tackled by performing Unsupervised Domain Adaptation (UDA), where a model is adapted to an unlabeled target domain by leveraging the labelled source domain. In medical scenarios, the data comes with huge privacy concerns making it difficult to apply standard UDA techniques. Hence, a closer clinical setting is Source-Free UDA (SFUDA), where we have access to source trained model but not the source data during adaptation. Methods trying to solve SFUDA typically address the domain shift using pseudo-label based self-training techniques. However due to domain shift, these pseudo-labels are usually of high entropy and denoising them still does not make them perfect labels to supervise the model. Therefore, adapting the source model with noisy pseudo labels reduces its segmentation capability while addressing the domain shift. To this end, we propose a two-stage approach for source-free domain adaptive image segmentation: 1) Target-specific adaptation followed by 2) Task-specific adaptation. In the Stage-I, we focus on learning target-specific representation and generating high-quality pseudo labels by leveraging a proposed ensemble entropy minimization loss and selective voting strategy. In Stage II, we focus on improving segmentation performance by utilizing teacher-student self-training and augmentation-guided consistency loss, leveraging the pseudo labels obtained from Stage I. We evaluate our proposed method on both 2D fundus datasets and 3D MRI volumes across 7 different domain shifts where we achieve better performance than recent UDA and SF-UDA methods for medical image segmentation. Code is available at https://github.com/Vibashan/tt-sfuda.