What's likely is that the parasite was already in your system but at sub-clinical levels, kept at bay by the immune systems of fish that have adapted to it. Then you introduce a fish with an innate susceptibility and after a couple of life cycles (a month sounds about right) your tang has shed enough tomonts in to the system that when they start hatching the concentration of infective parasites rose high enough to start overwhelming your otherwise resistant fish. It's like locking a bunch of vaccinated children in a classroom every day with a few who aren't, then exposing them all to measles. Eventually even the vaccinated kids are at risk just from the constant exposure.
If you have the set up, I highly recommend the tank transfer method. Super easy on the fish so long as you control for ammonia, guaranteed effective to eliminate ich from your fish 100% in 12 days, and there's no risk of poisoning them by accident. After that you'll have another 8 weeks to observe them and treat them for whatever else you think is necessary.
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