My Lips Don't Close Naturally. What Procedure Will Fix my Mouth?

There are many reasons for having an open mouth an inability to close properly. This is often related to the maxilla and mandible relations.  Withotu a formal exam and more than likely a need for cephalometirc studies, it is a difficult question to answer.

Your appear to have a long face and you have identified  many  characteristics  of this condition. Orthognathic surgery with impaction osteotomy with model surgery is best. I think fillers with HA or fat or implant would make your face look very stylized and unnatural. 
Michael Eisemann M.D.
Plastic Surgeon
Houston, Texas

Lip implants will not fix this condition. It sounds like you know what needs to be done. Unfortunately the limited view photo you posted does not allow any assessment of the situation so there is no way to know what all your options are.

My response to your question/post does not represent formal medical advice or constitute a doctor patient relationship. You need to consult with i.e. personally see a board certified plastic surgeon in order to receive a formal evaluation and develop a doctor patient relationship.

Save your funds for possible  "gummy smile surgery"  which should not be that much more expensive than lip implants. The other problems you describe may be costly to fix. You should be evaluated by an experienced oral and maxillofacial surgeon.