uh...I've never heard of this before. I've gotten god knows how much oil, solvent, paint etc. all over me and never run into this. 
If you lay on your back and lift weights - like bench presses - do you get the same thing?
I'm interested to see if anyone else has had this experience.
New deodorant ?
Never had a problem like this either and I've been elbow-deep in all kinds of automotive chemicals. Maybe a bad reaction to deodorant being brought about by more sweating than usual?
I had chemical burns from gasoline mixing with deodorant in my armpits a couple of times.
That's all i have for you.
I'm no doctor but allergies can make lymph nodes swell. Sticking your hands in a pile of petroleum distillates and other toxic chemicals that are absorbed thru the skin seems like a reasonable place to start looking for reasons.
When I get JP-8 in my armpits they do that.
Accumulation of fluid in your lymph nodes. You may have a circulation problem. Or you could have a kidney problem - not expelling fluid quickly enough so it accumulates in your body. Do your ankles swell when you stand or sit for a long time? My wife's an internal medicine doctor so I hear about stuff like this all the time.
Its worth looking into further.
Look at that, GRM really does have the answer for anything!
Anything involving swollen lymph nodes would make me set up a doctor's appointment asamfp.
You absorb more chemicals through your skin then through your mouth. Much more direct shot to your bloodstream too.
Wear gloves. You'll get used to it.
I've been a professional technician for 23 years and my hands look nicer than most desk jockeys. Protect your skin. Your body will appreciate it and your lady friends will appreciate it too. 
Trust me.
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