Droughts and heat waves have important impacts on multiple sectors including water resources, agriculture, electricity generation, and public health, so it is important to understand how they will be affected by climate change. However, there is large uncertainty in the projected changes of these extreme events from climate models. We compare historical biases in models against their future projections to understand and attempt to constrain these uncertainties. Historical biases in precipitation, near-surface air temperature, evapotranspiration, and a land-atmospheric coupling metric are calculated for 24 models from the Coupled Model Intercomparison Project Phase 5 (CMIP5) against the North American Land Data Assimilation System Phase 2 (NLDAS-2) as reference for 1979-2005. Biases are highly correlated across variables, with some models being hotter and drier, and others wetter and cooler. Models that overestimate summer precipitation project larger increases in precipitation, evapotranspiration, and land-atmospheric coupling over important agricultural regions by the end of the 21st century (2070-2099) under RCP8.5, although the percentage variance explained is low. Changes in the characteristics of droughts and heat waves are calculated and linked to historical biases in precipitation and temperature. A method to constrain uncertainty by ranking models based on historical performance is discussed but the rankings differ widely depending on the variable considered. Despite the large uncertainty that remains in the magnitude of the changes, there is consensus amongst models that droughts and heat waves will increase in multiple regions in the US by the end of the 21st century unless climate mitigation actions are taken.