Often there is substantial uncertainty in the selection of confounders when estimating the association between an exposure and health. We define this type of uncertainty as `adjustment uncertainty'. We propose a general statistical framework for handling adjustment uncertainty in exposure effect estimation for a large number of confounders, we describe a specific implementation, and we develop associated visualization tools. Theoretical results and simulation studies show that the proposed method provides consistent estimators of the exposure effect and its variance. We also show that, when the goal is to estimate an exposure effect accounting for adjustment uncertainty, Bayesian model averaging with posterior model probabilities approximated using information criteria can fail to estimate the exposure effect and can over- or underestimate its variance. We compare our approach to Bayesian model averaging using time series data on levels of fine particulate matter and mortality.