I found what might be the "HOLY GRAIL" of stain free deodorants and a method that fixes the problem!!
I have spent so much money on new work shirts, new casual shirts, and new undershirts. Finally, a friend of mine who had the same problem did several experiments with different products, and figured out a combination.
First (and foremost), before you wear a shirt, wash it and use vinegar in the washing cycle and never use either Liquid OR Dryer sheets for softener on the shirts that have issues. Predominantly, shirts that contain cotton will have the most issues, since cotton fiber is like velcro, and everything clings to it, and any sort of softener only makes it 'stickier'.
Then, right after you get out of the shower, cool your body off then squeeze a lemon wedge around the entire area surrounding your armpit. Let it set for a couple minutes then completely dry the underarm area THOROUGHLY..
The deodorant that does not cake up on cotton or materials is called "Herbal Clear". I use "Sport" fragrance or sometimes "Outdoor", because it smells really good like the major name ones. It contains Tea Tree Oil, so it also helps with any minor irritations, and says it's Hypoallergenic. The company is out of Tempe, AZ, and the back of the stick has the website: 21stcenturyvitamins.com , but I buy it at WinCo Foods, Trader Joe's or Albertson's. WinCo sells it for about $3.50, but TJ's or Albertson's sell it for $5, so I always stock up at WinCo. I noticed that it comes in 4 fragrances and one fragrance free.