In this work, we focus on improving the accuracy-variance trade-off for state-of-the-art differentially private machine learning (DP ML) methods. First, we design a general framework that uses aggregates of intermediate checkpoints \emph{during training} to increase the accuracy of DP ML techniques. Specifically, we demonstrate that training over aggregates can provide significant gains in prediction accuracy over the existing state-of-the-art for StackOverflow, CIFAR10 and CIFAR100 datasets. For instance, we improve the state-of-the-art DP StackOverflow accuracies to 22.74\% (+2.06\% relative) for $ε=8.2$, and 23.90\% (+2.09\%) for $ε=18.9$. Furthermore, these gains magnify in settings with periodically varying training data distributions. We also demonstrate that our methods achieve relative improvements of 0.54\% and 62.6\% in terms of utility and variance, on a proprietary, production-grade pCVR task. Lastly, we initiate an exploration into estimating the uncertainty (variance) that DP noise adds in the predictions of DP ML models. We prove that, under standard assumptions on the loss function, the sample variance from last few checkpoints provides a good approximation of the variance of the final model of a DP run. Empirically, we show that the last few checkpoints can provide a reasonable lower bound for the variance of a converged DP model. Crucially, all the methods proposed in this paper operate on \emph{a single training run} of the DP ML technique, thus incurring no additional privacy cost.