Do I need a positive bacterial culture to have sepsis in the blood?

Not always. Sepsis may be cause by other pathogens such as viruses and fungus.
Sort of. Sepsis is usually due to bacterial infections, but not exclusively, fungal infections can also cause sepsis. A person may be septic and not have a positive blood culture due to vagaries of specimen collection, prior antibiotic treatment and infection with organisms that may be difficult to culture.