Should I be doing more for this burn? How much time will heal? Weeks? Months? Just had LHR to my abdomen.

I recommend getting a formal evaluation with a cosmetic dermatologist for after care. I recommend Silagen + Bioshield to promote healing. Any scarring can be improved with lasers and microneedling/prp. See an expert. Best, Dr. Emer

The center is offering you all they can.  They should pay for you to see a board certified dermatologist.  They may even have one that they work with.  You should be starting treatment for a burn.  I would not use neosporin, as some people are allergic to it and you don't want to complicate the situation. Drinking water won't make any difference.  Staying out of the sun is a good idea.