If my lower back suffered from a severe contusion almost six months ago because I hit my back really hard against a sharp object and I am still dealing with occasional severe pain from the injured area, will the injured area be able to fully recover?

It can. if tests like mri ruled out anatomical pathology then yes try pt .
Needs to be checked. If after 6 months your back is still hurting and the pain when it comes is severe,i will suggest it is about time you get it checked by your doctor as you may need imaging like MRI and may need Physical Therapy and appropriate treatment depending on the results of exam and imaging.