How would we test for significant changes in expression when many of the normalised counts are clustered together?
TLDR- just do the analysis in DESeq2 (it can handle multiple covariates)
If I've interpreted your question correctly there seem to be a few confusions here.
First off, for a multiple linear regression aka ANCOVA, one of the assumptions are the residuals are normally distributed - however this is almost never an issue in practice. The distribution of the data is not so important, you make the assumption that the data has come from a population which is continuous and normally distributed that's why you've chosen to do an ANCOVA. But, RNAseq produces count data - not continuous and bounded at zero. You need to be looking at using a Poisson GLM, or negative binomial GLM as done by DESeq2 (lookup overdispersion)
The normalisation done by DESeq is to account for different library sizes,it doesn't transform the data to make it conform to a normal distribution. You'd need to do some form of data transformation to do that, but don't it's unnecessary.
I'd suggest reading the DESeq vignettes and following their standard pipeline.
Hope that is of some help.