Would a dna appliance or Invisalign work for me or would I really need braces? (photo)

Maxillary expansion is ideal for your treatment.  It is unfortunate that expansion was not done much earlier.  Ideally 7-8 years of age at the latest.
Braces / Invisalign will do nothing to correct your airway issues.  Expansiion Orthopedics will create a full wide smile and a far healthier airway in a process called pneumopedics. 
I assume you have a forward head posture already.  
Ira L Shapira DDS, D,ABDSM, D,AAIPM, FICCMO, MICCMOPast Chair, Alliance of TMD Organizations    Diplomat,  Academy of Integrative Pain ManagementDiplomate, American Board of Dental Sleep MedicineDiplomate, American Board Sleep and Breathing
Regent, Master & Fellow, International College of CranioMandibular Orthopedics
Board Eligible, American Academy of CranioFacial Pain
Dental Section Editor, Sleep & Health JournalCranioFacial Pain Section Editor, CRANIO: Journal of Craniomandibular and Sleep Practice
Member, American Equilibration Society
Member, Academy of Applied Myofunctional SciencesMember, Academy of Cosmetic DentistryLife Member, American Dental Association

It is hard to tell exactly what would be involved in fixing your malocclusion since you only provided a single photo.  Invisalign can align your teeth but may not correct your bite.  The fact that your midlines are off indicates a lot of tooth movement is necessary for full correction.  The DNA appliance should not even be considered in this case.  It does not provide 3 dimensional control of the teeth.  Braces would be best, Invisalign probably acceptable with a very experienced orthodontist supervising. The size of you maxilla is fine.  There is no such diagnosis as "underdeveloped maxilla".  Using a made up orthodontic sounding term to justify any type of orthodontic treatment  would cause me to look elsewhere for treatment.

basic difference between DNA and invisalign is, DNA is for dental expansion , on other hand invisalign/ braces is for full correction. as yr teeth in cross bite and canine is bucally places, invisalign with distalization will work best. invisalign technology is far ahead thn DNA appliance. much expansion is also not recommonded, that increases relapse tendency.