Okay thanks for clarifying. This would also indicate that you were not expose to Chlorine Dioxide per say, but more accurately Sodium Chlorite.
That said, I'm not well versed on the chemical interactions of Sodium Chlorite beyond what I've read on ingestion studies in humans. However, since you didn't actually swallow this, I'd say it isn't likely that you would have reached any form of toxicity given the very short exposure times(under 1 min) and the very small quantities that would have passed transdermally.
For what it's worth, I've handled Sodium Chlorite in both powder and liquid form with my bare hands on numerous occasions over the years without any side effects beyond that of dry skin afterwards. Which is likely due to the Sodium component.
Having said all that, the main issue with Sodium Chlorite ingestion seems to revolve around renal failure. Though it's also worth stating the quantities needed to reach toxicity are proportionate to ingestion. And so I'd say you have little to worry about beyond that of the effects of the Sodium in your mouth and lips.
Beyond that, I highly doubt you could have caused any DNA damage in doing this. If anything, your mouth reacted in much the same way it would have, if you swished with a concentrated table salt solution. :)
Hope this helps.
PS. You need to be very careful when handling products like Miracle-Mineral-Supplement (Sodium Chlorite) as mishaps due to confusion and/or inattention can so easily occur. ie, whenever I handle chemicals, I make a point of using dedicated flasks that I label with markers at the time of use. This really helps avoid any confusion during handling, after which I simply wash off the marker afterwards.