[CK] Not normal in Shapiro but normal in QQ plot

You do not have a problem here. Your data my be slightly non-normal, but it is normal enough that it shouldn't pose any problems. Many researchers do statistical tests assuming normality with far less normaldata than those that you have.

I would trust your eyes. The density and Q-Q plots look reasonable, despite some slight positive skew on the tails. In my opinion, you do not need to worry about non-normality for these data.

You have an N of about 350, and p-values are very dependent on sample sizes. With a large sample, almost anything can be significant. This has been discussed here.

There are some incredible answers on this very popular post that basically comes to the conclusion that conducting a null-hypothesis significance test for non-normality is "essentially useless." The accepted answer on that post is a fabulous demonstration that, even when data were generated from a nearly Gaussian process, a high enough sample size makes the non-normal test significant.