The puffiness is actually the fluid and soft tissue build up that occurs with the onset of gyno. This is the time that AIs are going to be most effective, because once you get the hard fiberous lump, you are 99% stuck with that and only surgery as an option to get rid of it.
You can try to have insurance to pay. Everyone that I know that has had gyno from puberty had their covered by insurance. But this isn't to say that you run down to the doc and just claim it was puberty related. They actually had doctor visits back in their early teens to document the development.
If you are patient, what you can do to increase the chance of insurance paying is go to the doctor now and tell him the reason for your visit is you have these lumps that came on in the past couple of months. That they worry you. Nothing else. He will give you the standard spiel, etc.. Go back in about 4 months and tell him that they are starting to hurt and feel similiar to a bruise. Again, nod your head at the speech. Wait a while longer go back again and tell him the pain is intense and hurting and you can sleep on your stomach or do things like hold your kids up against your chest or wear your selt belt. After this he will put it through to the insurance company as something other than cosmetic surgery.
Or if you live in body beautiful city - south florida, CA, NYC, etc.. - you can probably just go straight to a doc that will already know what to do to make insurance companies pay right off the bat and pretty much tell you what to put down on the visitation form to justify it.