Jet Travis

Does this chamois thing need a break-in period?
No, the chamois should not need to be broken in. It should not be stiff nor should anything bunch up.

Are they available thinner and more flexible? Or smaller?
Chamois come in all sorts of configurations. And so do shorts. You'll spend more for "eight panel" shorts" than for six panel or four panel. But they fit your body better, especially when you are in a riding position.

Can you get shorts without one entirely?
Yes. I think they're called compression shorts. You probably won't find them in most bike shops.

Are the issues I'm having due to buying cheap, generic shorts?
Very possible. My bet is that if you went up a couple of price points, you'd have better success. We have a Performance shop in town. I've been happy with their mid-range stuff (which often goes on sale). Pearl Izumi is a solid name, among many others. If you'd like to spend a mortgage payment, check out Assos (too rich for my blood but considered excellent by many).

Is there something I just don't get?
Maybe you're just comfortable riding without bike shorts. Nothing wrong with that. People have toured accross continents in gym shorts and sneakers. I rode for quite awhile without bike shorts and didn't seem to suffer any ill consequences. But once I experienced the difference, I stayed with cycling-specific shorts and never looked back.