I'm 5'5" and weigh 120 lbs. I'm a 34DD and would love to be at most a 34C. Is this something insurance would cover? (Photo)

Thank you for the question and pictures. Based on your description, you may be a good candidate for breast reduction surgery. “Official” recommendations of course would necessitate in-person consultation.
Breast reduction surgery is one of the most patient pleasing operations we perform. By removing “excess” breast tissue, adipose tissue, and breast skin  this operation reduces AND lifts the breasts to a higher position on the chest wall (in other words, the breasts are "automatically" lifted when a breast reduction is performed). By doing so, patients often find improvement in neck, back, and shoulder discomfort and find it easier to form their activities of daily living and exercise.
Timing of the operation will depend on the patient's life circumstances. Sometimes breast  reduction surgery is covered through health insurance. The best way to obtain insurance coverage for breast reduction surgery involves some “hoops” to jump through. The more documentation you have (for example, from your primary care doctor, physical therapist, chiropractor etc.) the better when it comes to obtaining insurance “authorization” for the procedure.
 This documentation and letter/pictures from your plastic surgeon will help you obtain authorization.
When the time is right,  seek consultation with well-trained/experienced board-certified plastic surgeons. Ask to see lots of examples of their work and communicate your goals clearly. Best NOT  to communicate your goals in terms of achieving a specific cup size; this type of communication can be a source of miscommunication. In my practice, I prefer the use's of goal pictures during this communication phase;  many patients wish to remove enough breast tissue to improve/alleviate symptoms while maintaining upper/lower body proportionality.

I hope this, and the attached link (dedicated to breast reduction surgery concerns), helps. Best wishes.

Your photos would indicate that you could really benefit from a full breast reduction and lift. Seek the top breast plastic surgeon in your area for this.