I think nausea counts as morning sickness! I didn't throw up much with this pregnancy (maybe 11, 12 times), but the nausea was debilitating and constant. I definitely considered it morning sickness, even if I wasn't throwing up most of the time.
Unisom/B6 worked for me and I was able to come off it at around 12 weeks with no issues. To some extent, mint gum and those pregnancy drops with B6 also helped. Also just bland, starchy foods -- the one I liked was Rice Krispies, saltines grossed me out.