I have lip incompetence. What is the best way to fix this? (Photos)

I would highly recommend getting a formal evaluation with and treatment from a board-certified cosmetic dermatologist and expert in facial sculpting. Technique is the key to a successful lip augmentation with fillers, so you'll want to see someone meticulous. An experienced physician will be able to determine which filler, how much filler, and which technique will give you the subtle result you're looking for. See an expert. Best, Dr. Emer

After looking at your pictures, a lip augmentation may not be the best pathway for you. The same opinion for your feelings about a chin augmentation. 
You would be best served by seeing a maxillo-facial surgeon.  Your breathing problems may not be related to your lip position.
Best, Dr. Kulick

Lip incompetence can be a function of a shorter chin or an elongated upper jaw and in some cases both. You would need to gave am orthodontic/orthognathic surgery evaluation to determine whether either of these surgeries could adequately address your lip incompetence concerns.

You need to be evaluated by a plastic surgeon or a Maxillofacial Surgeon for maxilla reduction surgery. Good luck from New York City

This may not be the best place to look for information on your condition. I do not think treatments like Botox or filler‘s are going to help you with this.I would recommend getting evaluated by ear nose and throat specialist or oral surgeons. Call around and look for people who have experience with your condition.Consider going to an academic medical center.Best, Mats Hagstrom MD