Determining genetic variant effects on molecular phenotypes like gene expression is a task of paramount importance to medical genetics. DNA convolutional neural networks (CNNs) attain state-of-the-art performance at predicting variant effects on gene regulation. However, most applications of such models focus on single nucleotide polymorphisms (SNPs), as technical challenges limit their application to insertions and deletions (indels). Sequence shifts from indels introduce technical variance in deep CNNs through misalignment of pooling blocks and output boundaries, creating artificially inflated variant effect scores compared to SNPs and confounding their interpretation. In this work, we demonstrate this technical variance in model predictions and present two strategies based on data augmentation with sequence shifts that reduce it. Applied to the state-of-the-art Borzoi model, our stitching approach improves indel eQTL classification accuracy across GTEx tissues. Furthermore, we demonstrate these techniques and observe compelling eQTL concordance for larger structural variants and tandem repeats. We additionally introduce in silico deletion (ISD) as an interpretation technique and validate it using MPRA data, demonstrating concordance between predicted and experimental measurements for deletion effects. Our strategies expand the utility of regulatory sequence machine learning for studying the full spectrum of noncoding genetic variation in human development and disease.