Residential heating, primarily powered by natural gas, accounts for a significant portion of residential sector energy use and carbon emissions in many parts of the world. Hence, there is a push towards decarbonizing residential heating by transitioning to energy-efficient heat pumps powered by an increasingly greener and less carbon-intensive electric grid. However, such a transition will add additional load to the electric grid triggering infrastructure upgrades, and subsequently erode the customer base using the gas distribution network. Utilities want to guide these transition efforts to ensure a phased decommissioning of the gas network and deferred electric grid infrastructure upgrades while achieving carbon reduction goals. To facilitate such a transition, we present a network-aware optimization framework for decarbonizing residential heating at city scale with an objective to maximize carbon reduction under budgetary constraints. Our approach operates on a graph representation of the gas network topology to compute the cost of transitioning and select neighborhoods for transition. We further extend our approach to explicitly incorporate equity and ensure an equitable distribution of benefits across different socioeconomic groups. We apply our framework to a city in the New England region of the U.S., using real-world gas usage, electric usage, and grid infrastructure data. We show that our network-aware strategy achieves 55% higher carbon reductions than prior network-oblivious work under the same budget. Our equity-aware strategy achieves an equitable outcome while preserving the carbon reduction benefits of the network-aware strategy.