I stopped using antiperspirant nearly 2 years ago but I didn't go through anything with cysts so I'm not sure about that being a detox issue? You may be right that it's a reaction to what you started using in place of your regular deodorant so switching it up may be a good idea or going without to see if it clears up.
When I switched from using antiperspirant to natural I was a little stinky for a week or two while while my skin got "unclogged" from the gunk that seems to get locked in from not being able to sweat it out.
Now I use an all-natural cream deodorant from Bubble & Bee called Pit Putty.
It's been very good but it's definitely different than using a store bought antiperspirant so don't expect it to be the same. It does not stop you from sweating and you may need to put it on more than once a day if you are very active or live where it's warmer. We get so used to something that is supposed to "protect" us from odor all day that it's a step back using something natural but I feel so much better about putting this on my body that I don't mind a little extra maintenance to be healthier in the long run.