I agree, I have been wearing a athletic bra most of the time now the past few weeks, and when I really want to be 100% not seen, I wear a compression shirt or similar over it. But not detectable unless touched to a point of exploration or under the arms where it would be noticed more so. Heck I look and feel better with it all. You can deal with heat much better than you think. Women do it daily in all weather. They survive fine.
Even hot and sticky out it is far more comfortable wearing than not wearing, seeing a sweat line under my breasts all the time and the ick feeling of skin on skin and from time to time sores is no more. Sure, I want to get it off later, but still more comfortable on.
I have a XL and L ath bra and the XL fits a bit better, the "cups" if you will are a tad larger, but the L is tighter, which by the end of the day is more annoying.
You probably need to see a dr, and you need to be talking to your parents about it, mom most likely in private. You will find them to be more open than you realize. They will want to help, not judge. She will help you decide what is better to take care of it. But if they don't know how you feel and what you need, no one can help.
I will say it again.
An athletic bra should give the support you want without making a spectacle of you. A "regular" bra will tend to cause the breast to be somewhat more elevated and projected, therefor much more obvious to anyone who sees you.
The various compression garments should be tried first. The are intended to compress and support making the breasts much less evident.
If you do not know of a Doctor nearby it should not be a problem as your parents no doubt do know of one.
If your condition is as you have described, it should be obvious to your parents and they have avoided saying anything in an attempt to avoid embarrassing you. You will need to speak up and let it be known that you have a need to see a doctor.
I was not merely trying to be dramatic; if your condition is as you said your general health and development should be reviewed by a medical professional. This can only be done with your parents approval.