Both seem to mean contest, resists. Is あらそう more about verbal protest? And あらがう related to action such as rebellion? Is there an overlap?
There is a slight difference.
「あらそう」 is to fight with, to compete for something, http://ejje.weblio.jp/content/%E4%BA%89%E3%81%86
「あらがう」 is to oppose, to resist, to go against something http://ejje.weblio.jp/content/%E3%81%82%E3%82%89%E3%81%8C%E3%81%86
Simply put, you can 「あらそう」 for 1st place, but not 「あらがう」. On the other hand, as a soldier, you can 「あらがう」 a direct order from your superior if you find it immoral, but its's not something you can 「あらそう」.
Either word can be carried out verbally or via action, so that wouldn't be the defining difference.