Hmmm...it sounds to me like you are not shaving correctly. I would try these steps.

Shower, with warm water in your 'pit' (he he he). Wait a minute or two, then, lather well with shave creme (frankly, I think they are all the same, the women's stuff just costs more, IMO). Then, with a fresh razor, shave the area from the top (closest to your elbow) down (closest to the body). BUT make sure your arm is up over your head, with the largest possible exposed surface, as flat as you can make it.

THEN, when you get out of the shower, pat yourself dry there and DO NOT put on deoderant for as long as you can go without....maybe right before you walk out the door??? (Do you even NEED deoderant? LOL..that might be another thread...I don't wear it...I know, I know, TMI! But, I don't NEED it (yes, everyone agrees with me....he he he))

Good luck. Also, I always found it easier on my skin to shave every day, not skipping any days. It was only when I started shaving 1-2 times a week that I would get cuts or burn.