Just finished antibiotics for my tooth abscess, but I am continuing to have pain and swelling. Is it still infected?

Yes. The infection is still present. You may need to have a different antibiotic prescribed if more definitive treatment, i.e., root canal therapy if indicated, removal of the tooth and/or incision and drainage of the abscess if the tooth is non-restorable and drainage is not established through the extraction site.
Most likely. Not all abscesses are the same. If you are not responding to the medication, your doctor may have to try a different one. Unfortunately, there are many drug resistant bacteria and finding the right medication is a process. Understand that medication alone is not going to solve your problem. Make sure you also have the appropriate dental treatment completed.
Likely. The abscess might need to be cleaned out by the dentist.