Is there any way besides surgery to get rid of my puffy nipples? (photos)

Thank you for your question. Given the timing of your problem - and its duration - it appears as though you have a mild case of gynecomastia. Gynecomastia is when males form breast tissue - it can present directly behind the nipple areola complex (which it looks like in your case), or can actually encompass the entire breast. Although there can be other causes of gynecomastia, including precocious puberty, hormonal imbalances, medication-induced, etc., most cases are idopathic - meaning, we're not quite sure why they happen. Your likely best bet would be to see a Plastic Surgeon for a full head-to-toe examination, and to see what your options are. Given your body frame, and the relative lack of breast tissue, this tissue may be able to be liposuctioned - or, if it's too fibrous, it can be directly removed though a semi periareolar incision. Best of luck!