Early on, I shopped Target, Kohls and Old Navy frequently. Kohls is easy, just peruse the womens sections then drop it in the black bag, go into the mens changing room and have at it, no one (except maybe the burly dude monitoring the security cameras, whom I imagine with a box of goopy donuts next to him). Old Navy, kind of the same thing. Target, I never wandered into the changing room but have found the fit pretty consistent, so once you know your size, you can somewhat reliably pick through jeans, t-shirts, etc.
I tried the whole amazon thing but always ended up disappointed because the fit was all over the map and I'd end up with multiple exchanges for size, which is always a let down.
I was never really embarrassed or shameful shopping in the women's department, mostly because I do it all the time anyways for my girlfriend/wife/etc. So what's the difference if it's for me or someone else? only what's going through my mind. My money is still green, so no one really cares anyways.