It's simply a matter of reframing. Admittedly, the first few times, it's going to be embarrassing no matter what, but that's only because we're viewing the situation from our own biased perspective: WE know the diapers are for us and WE know what we'll be using the diapers for and that's how WE are inevitably seeing the transaction. It takes a little while until we can finally realize that not only do THEY not know the diapers are for us and THEY don't know what we'll be using the diapers for, they honestly don't care.
One of my first jobs was working as a cashier at Target and that really helped me when it came to buying in public. I'd scan thousands of items per day, and adult diapers were regularly one of them. Of course *I* noticed it because that's what I'm into, but if I wasn't into it, I'd have paid it no mind whatsoever. Let's say for example that somebody had an enema fetish - gross (to me), I know, but maybe they have the same feelings of embarrassment having to buy them at the store. If you were to ask me if I sold any enemas and to whom I sold them, I couldn't answer because I'm sure I did, but as I don't ever think about enemas, it never stuck out to me as I sold them. It's the same thing with diapers because the general public doesn't think about them like we do. It's just one of thousands of products they sell each day: your particular pack of diapers won't be their first and it won't be their last.
I've bought so many diapers at the store over the years - including trips where I bought only a pack of diapers - and have never had any sort of weird look or comment, mostly because if they make fun of you for any purchase (ESPECIALLY sensitive healthcare items), they would be fired. I guarantee the only thing going through their minds as they check you out is just wondering how long it is until their lunch break. Once you look at the situation from their perspective, it gets so much easier to buy in public.