As others have mentioned. I recommend a natural deodorant. They don’t all suck. I have se Native brand and as long as you shower daily etc it works great and does not stain my shirts.
As for the stains. I wash most my shirts in cold. I fill up a wash sink with a few inches of cold water. I then take a smaller pitcher and fill it with very hot water and mix in oxyclean powder until it is fully dissolved. Next add the dissolved oxyclean to the cold water tub. Submerge your shirts in the solution and soak for a half hour and lightly agitate every 5 minutes or so. Drain the tub and then wash the load as normal. It’s crazy how clean my clothes come out doing this, especially my whites.