I'm wondering if maybe it's not razor burn since you say you've always had the bumps. With razor burn the bumps usually go away after a few days. You could have another skin condition like keratosis or foliculitis. Keratosis is caused by too much keratin in the skin and foliculitis is inflmation of the folicles and can be cleared up with medications.
Definitely use shaving gel for sensitive skin or body wash, never soap. Also underarm hair grows in every direction so there is no "grain" per say. You may want to try shaving up or down from left to right like a zig zag, or sideways instead of straight up and down. Also apply lotion after shaving and don't apply deoderant right away if you can help it. It also could be your deoderant even though you don't have any other skin allergies. You may want to try a deoderant for sensitive skin. I know Dove works great for your skin, but isn't real great for odor control so don't use Dove before working out lol!
Let us know if you find something that workss!