[wilcoxon/paired t test] when to exclude data? Not normally distributed data.

Hello! I ran an analysis of my data of research into a repeated measures design and found the difference of both conditions to be not normally distributed.

When exlcuding data to make it normally distributed and thus use parametric tests, can we only exclude data that exceeds the limit of the z score +-3.29 (outliers), or can we exclude data that exceeds the 5% of data that goes up to Z score of +-2.58.

For an example, say 8% of my data goes up to +-2.58. Am I able to exclude the 3% so only 5% of my data is within the limits of +-2.58, or can I only exclude outliers (none)?

Thank you!