Is a dentist suppose to start the process of performing a root canal without draining the abscess if it's swollen the cheeks?

Yes, of coarse: The only way to cure a dental abscess is with dental treatment. The treatments are: root canal treatment, tooth extraction and/or in some cases, drainage surgery. The treatment for this type of abscess is normally root canal treatment. Hope it helps.

Swollen abcess: Depending on the size of the infection and the amount of sweliing, we will sometimes place a patient on antibiotics for several days prior to starting a root canal. This usually depends on whether the treating dentist feels antibiotics are necessary.