You can use flax in place of the eggs; I wouldn’t add both.

I've mixed ground flax with water to make vegan eggs for baking. It works out great for muffins. For cookies, the texture was off but they still tasted good. Check out Oh She Glows. It's not operational anymore but all the recipes are still posted. All vegan and incorporates a lot of flax and chia. She has some cookie recipes. Everything I've made from there has been good, although sometimes a hassle since I am not vegan and could easily use eggs, butter...etc.

I’d like to add some nutritional benefits to my oatmeal cookies by adding flax.
I’ll have nuts in these too but they won’t affect how dough binds.

Would you recommend adding addition moisture?

Would you do so by soaking the flax? Adding more a bit butter/oil/fat? Add milk? Add egg yolk?

Baking has chemistry going on! I don’t want to waste ingredients (and time). If you have any advise please share.

Or a tried & true recipe could you please share? Googling is just a endless scroll of blogs.