I have a regression equation whose residuals are homoscedastic but have a slight negative skew due to several outlier values. The Q-Q plot is pretty close as to what would be expected from a normal distribution, but has a slight deviation in the upper quantile that seems to be related to these values. I am trying to test whether these outlier values cause the residuals of the regression to depart from normality enough that it is inappropriate to use the regression equation to predict the values of new data.
I originally tested for normality using a Kolmogorov-Smirnov test using the estimates mean and standard deviation of the sample as the comparative sample and that said the results did not depart from near normality, but as I read more about statistics I heard several people recommend that the Kolmogorov-Smirnov test should never be used. So I tried a Shapiro-Wilk test instead and the result came back as significant.
However, I suspect the reason the Shapiro-Wilk test produced a significant result was because of the large size (~350) of the dataset, as I've read that at larger sample sizes even small deviations from normality will cause the Shapiro-Wilk test to give a significant result, even if the data approximate normality. This is supported by the fact that even when these outlier values are removed the Shapiro-Wilk test still finds the residuals to significantly depart from normality. This is the reason I originally used a Kolmogorov-Smirnov test, because it is supposedly more robust to large sample sizes.
I realize my sample size is probably large enough that I don't have to be concerned about non-normality, but it would be nice to find a better way to say that the residuals are close enough to normal to work than simply invoking the Central Limit Theorem. Given this, what would be the best way to test if my data are "nearly normal"?