Bangladesh is one of the climate risk-prone countries in South Asia facing tremendous challenges to combat sea-level rise and its associated coastal morphological changes. This study aimed to determine the interaction of the sea-level rise and morphological changes, particularly at Cox’s Bazar and Kutubdia Island along the eastern coast of Bangladesh. Available hourly tide gauge data, daily temperature, daily rainfall data, and 15 LANDSAT satellite images for the period of 1983–2016 were analyzed to examine the sea level shore morphological change and associated climate change phenomenon. First, we identified the historical nonlinear sea-level trend using Hilbert-Huang Transformation (HHT) based on the complete ensemble empirical mode decomposition (CEEMD) technique. We divided the study period into three distinct sea-level change periods of 1983–1993, 1993–2003, and 2003–2014 based on nonlinear sea-level trend analysis. The study revealed that the sea level on the east coast of Bangladesh had a moderate rising trend during 1983–1993, slight decrease during 1993–2003, and steep rising trend during 2003–2014. We also observed that a sea-level change within a particular period impacted the shore morphological change after approximately two years, such that the average sea-level change during the period of 1993–2003 might have affected the shore morphology for 1996–2005. Alarming shore erosion was found for the period of 2005–2016 compared to the previous periods of 1989–1996 and 1996–2005 for both Cox’s Bazar and Kutubdia Island. The shore morphology of some segments was also substantially affected due to the geometric shape of the land, significant waves, and shore protection works. This study encourages policymakers to minimize the threats of sea-level rise and ensure sustainable coastal management strategies are introduced to sustain the vital eastern coast of Bangladesh.