I don't know if Bali brands are available in RSA, but they also offer at least
one style in "smart sizing" that is much more forgiving.
Sizing chart:

You can buy this brand and model at retail or online. And it comes in plus sizes if you're a bigger girl.
Probably not a big favorite with women who need serious support, but it should work for you at this stage and it's at least a
little prettier than a sports bra, though not what I'd favor -- very, very plain, and I tend to find that underwires work better for me at this point, since they hold breastforms in place more reliably than other styles. It doesn't help your confidence if your breasts start migrating after an hour or two and you find yourself frequently leaving to put them back in place.
As others have pointed out, developing a more confident and matter-of-fact attitude helps a lot. Easier to use a changing room without negotiation if you shop somewhere with men's and women's changing rooms. Maybe it's different where you live, but I found that many women's shops are very welcoming if they think it will mean a sale, and will have no problem offering you a changing room as long as you don't act overly nervous and are consistently polite and respectful in your requests.
And bigger stores are often a good choice if you just don't want to talk to sales staff. I've shopped Kohl's a lot lately, and have had no one even seem to notice when I took fairly large piles from the women's departments to the men's changing rooms. Maybe it helps a little that I'm already wearing what I suspect the female staff, at least, recognize as women's apparel, though I'd think that would possibly get me noticed more, not less. Since I've been dressing this way for about 20 years, I guess the fact that I don't think much about it also means that others don't read me as nervous and start looking for the reason why. Smiling also does wonders to reassure people that you are just there to shop. This works even if you're practically certain someone has clocked you.
But it seems to me, in general, that their staff is mainly too busy to notice. I also live in a suburban area between two fairly cosmopolitan cities where staff have probably been trained not to bother customers except when they ask for help.