Whenever i run overwatch and stream with OBS, my framerates go from 300 on low setttings to 100 and they continually go down unless i restart overwatch or stop streaming entirely. Im running an i7 7700, a gtx 1070, and 16gb of ram. Havent tried switching from OBS, because it would be a hassle trying to reset it again.
Should try searching for guides regarding CPU Priority and Affinity for OBS.