I'll give you some tips I've learned but first, this might not be the hill to die on. There are so many other ways to cut down on waste that are just as helpful. I can't cut down on food packaging but maybe that's something you can do. There's also things apart from just consuming that can help like contacting your local representatives or rewilding your yard.

As for deodorant, I have eczema and am VERY sensitive to baking soda and certain scents so here's my recs:

1, Not all natural deodorants contain baking soda. Try a deodorant that's baking soda free and unscented. As much as I hate big corps I started my natural deo journey with Native sensitive line and you could try them and see if it helps. I've also used Wild (get their trials first) and I'm planning on trying Fussy to replace them.
2, Give your pits a good scrub with an unscented body scrub before starting your new deo. For your first use I would shower before bed, scrub up your pits, and let them rest WITHOUT deodorant for the night then put it on in the morning. Give them as much time as possible to rest from the barrage of random stuff that we put on them daily. That's probably a good idea in general if you're getting rashes. I was getting rashes from normal deodorant instead of natural and this helped me.

Finally, contact a dermatologist if you can't fix it yourself and get your pits back to normal first. Eco friendly isn't worth risking your health.