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Following a converter outage in a Multi-Terminal DC (MTDC) grid, it is critical that the healthy converter stations share the power mismatch/burden in a desirable way. A fixed value of power-voltage droop in the DC link voltage control loops can ensure proper distribution according to the converter ratings. Here a scheme for adapting the droop coefficients to share the burden according to the available headroom of each converter station is proposed. Advantage of this adaptive (variable) droop scheme for autonomous power sharing is established through transient simulations on an MTDC grid with four bipolar converters and DC cable network with metallic return. Results for both rectifier and inverter outages under two different scenarios are presented. Post-contingency steady-state operating points obtained from transient simulation are shown to be consistent with those derived analytically. Impact of varying droop coefficients on the stability of the MTDC grid is established. An averaged model in Matlab/SIMULINK which has been validated against detailed switched model in EMTDC/PSCAD is used for the stability and modal analysis.