Could diverticulosis cause severe stomach cramps?

Diverticulosis, no. Diverticulosis should not cause any symptoms you'd notice day to day. Diverticulitis, an infection of the diverticuli, could cause bad stomach cramps along with fever and diarrhea. If your symptoms are in between these two extremes, then maybe something else is causing them.
Not exactly. Though people with a long history of diverticulitis ( infections) may end up with scarring of the colon that leads to narrowing and severe cramping. Best to see surgeon or GI for colonoscopy. Make sure it isn't something else .