Blackheads result when a pore is only partially blocked. The black color is not caused by dirt. Rather, it is the skin's melanin reacting with the oxygen in the air. A blackhead can often take a longer time to clear than other types of acne.
With the other type of acne you're experiencing, are they nodules that are under the skin, or whiteheads, or pustules? Whiteheads and pustules are mild forms of acne, quicker to rid, but can leave some scarring if not treated with care. Nodules or cystic acne are severe types of acne and require agressive forms of prescription meds.
If you have the mild type of acne, then I'd suggest treating the skin externally and from within. Skin-friendly supplements, such as Vitamins A, Bs, C, D, and E, iron, fish oil, and zinc can help. Make sure not to exceed the recommended doses, as it can be toxic. Plenty of water throughout the day will also help your body rid toxins and keep your pores supple and clean.
Although your skin is oily, constantly washing away or using harsh treatments to dry out the natural sebum (skin's oil) will lead to an imbalance of the skin's pH, thus leading to acne or other skin conditions. Use a gentle facial cleanser and facial toner each morning and night, along with a moisturizer for sensitive skin. If you must extract a pimple, do so after cleansing the skin and taking a steam shower. If clear liquid or blood is extracted, stop, as this will lead to a scar.
If you use facial make-up, make sure the packaging specifically says non-acneic, non-comdegenic, or won't clog pores. Wash any make-up applicators at least once a week. And wash your pillow covers or any other item that may come in contact with your face at least once a week. All of these things can harbor bacteria that leads to acne. Good luck!! I hope you find clearer skin within the next few weeks of using this new regimen.