What can I do to fix/improve my lip incompetence? (Photos)

Typically problems like yours are related to breathing and tongue position.
Start treatment with an Oral Myofunctional Therapist who can then guide you to an educated airway dentist or orthodontist.

Do not let anyone amputate teeth but you can consider Epigenetic Orthopedics with a DNA Appliance.

Cece, I appreciate your concern and while physical exam is optimal, your photos demonstrate that you would most likely benefit from lip reduction surgery.
This would address your lip incompetence, correct it and also improve the shape of your lips by decreasing their overall size. The procedure is done in the office, under local anesthesia, and it takes about one hour. Recovery is usually one week, as postop, the lips are quite swollen. Seek out a board certified plastic surgeon with lots of experience with lip surgery, good luck!