Natural gas extraction from gas hydrate sediments by injection of hydrate inhibitors involves the decomposition of hydrates. The evolution of dissolved gas from the hydrate melt is an important step in the extraction process. Using classical molecular dynamics simulations, we study the evolution of dissolved methane from its hydrate melt in the presence of two thermodynamic hydrate inhibitors, NaCl and CH3OH. An increase in the concentration of hydrate inhibitors is found to promote the nucleation of methane nanobubbles in the hydrate melt. Whereas NaCl promotes bubble formation by enhancing the hydrophobic interaction between aqueous CH4 molecules, CH3OH molecules assist bubble formation by stabilizing CH4 bubble nuclei formed in the solution. The CH3OH molecules accumulate around the nuclei leading to a decrease in the surface tension at their interface with water. The nanobubbles formed are found to be highly dynamic with frequent exchange of CH4 molecules between the bubble and the surrounding liquid. A quantitative analysis of the dynamic behavior of the bubble is performed by introducing a unit step function whose value depends on the location of CH4 molecules with respect to the bubble. It is observed that an increase in the concentration of thermodynamic hydrate inhibitors reduces the exchange process, making the bubble less dynamic. It is also found that for a given concentration of the inhibitor, larger bubbles are less dynamic compared to smaller ones. The dependence of the dynamic nature of nanobubbles on bubble size and inhibitor concentration is correlated with the solubility of CH4 and the Laplace pressure within the bubble. The effect of CO2 on the formation of nanobubble in the CH4-CO2 mixed gas hydrate melt in the presence of inhibitors is also examined. The simulations show that the presence of CO2 molecules significantly reduces the induction time for methane nanobubble nucleation. The role of CO2 in the early nucleation of bubble is explained based on the interaction between the bubble and the dissolved CO2 molecules.