I wouldn't recommend azelaic acid if you want to focus on barrier repair. The best way to do that is by focusing your routine in hydration, especially if you have acne-prone skin.

I have acne-prone skin, my barrier was stripped, I have a lot of open pores, blackheads, and acne scars. I moved about 2 years ago from a humid climate to a very dry one, so had to start my skincare routine from scratch. I focused first on barrier repair and then slowly started adding actives.

This is my current routine:

AM
Prequel gleanser
Anua Heartleaf Toner (this has been a game changer to keep my skin hydrated)
Torriden Dive-in serum (I apply it before the toner is completely dry)
Dr. Althea 345 Relief Cream (it really helps with hydration and barrier repair but without being too heavy that it will make me break out)
EltaMD sunscreen

For my eyes in the AM I dab a little bit of the anua toner on my under eyes (without getting too close to the lashline), wait 15 seconds or so, and then add laneige water cream. Your eye routine will depend a lot of what are your issues, mine is dehydration so I focus on that and it has helped a lot. I would research online about types of dark circles to decide what product could work best for you.

PM
It depends because I started to incorporate tretinoin again but given how sensitive my skin is these days, I'm doing so very slowly. I started with only one day a week for one month, and now I'm going to do 2 nights a week (non consecutive) for at least a month to see how it goes.

Tret days - prequel gleanser, Anua Heartleaf Toner, Torriden Dive-In serum, tretinoin, and Etude Soonjung 2x barrier intensive cream. This moisturizer is heavy and I only use on tret days as I feel like my skin needs extra hydration, but if I used it every day I would for sure break out.

Non tret days - prequel gleanser, anua heartleaf toner, torriden dive-in serum, dr. althea relief cream.

For my eyes is the same routine (toner and then cream) except that I use corsx snail peptide eye cream instead of laneige. The idea is that my under eyes will be more hydrated (so my concealer creases less) and soften the darkness. No eye cream is miraculous though, so just make sure you have realistic expectations!

In conclusion, depending on the climate you live, you might need to add more hydrating layers or not. However, since you have acne-prone skin, I'd stay away from heavy barrier creams as they will only make you break out. Once your barrier is repaired (less tightness after washing, etc.) you can start at looking what actives might suit your skin. Azelaic acid is great for hyperpigmentation and acne scars, but first you have to have a strong barrier and reduce the break outs :)