Brief Answer:
Overactive sweat glands in the right armpit region could be a possibility...

Detailed Answer:
Hello and Welcome to 'Ask A Doctor' service...
I've gone through your query in details...
I can understand your concerns...

Well, if this has started happening only recently, then, first of all, please check whether the right arm is overactive/moving more then the left, try to cease some of those movements, while sitting put the right arm on the arm-rest/elevated to improve ventilation and observe...

If this is going on for a while now, then overactive sweat glands in the right armpit region could be a possibility...

Please wear a bit loose fitting clothing and start using a good quality Antiperspirant daily...along with it, good underarm hygiene (daily shower, application of body wash, routine waxing/shaving) should be practised...

Not to worry, as this is not something serious. Although if you feel there's still no improvements despite trying this, then there are other treatment options available (eg.stronger prescription based Antiperspirants)...There is even Electromagnetic therapy done by the Dermatologists and Plastic surgeons at their clinics where they permanently eliminate those over-active glands which are causing the excessive sweat production...

Let me know if I can assist you further.

Take Care
Kind Regards

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