I have very thick, medium (for a guy) length hair - just long enough to need a shower every day. If I don't, it gets pushed around by my pillow and won't go back into place until I wet it again. And when I wet my hair and don't use shampoo, it feels stiff.

If you feel that bad from a shower then it's entirely possible that you need to use a different shampoo/conditioner/soap/whatever. There are so many out there - some for dry hair to keep it oily, some for oily hair to keep it dry, etc. If you use oily hair shampoo when your hair is dry in the first place then obviously it's going to get even drier, so that's no good. I'm sure you can figure out what it is that you need to do.

Soap is a similar thing - some of them will dry you out more than others. But I would also suggest trying a slightly colder shower. I find hot water leaves me dry and lukewarm water doesn't. Of course, I personally don't care and just have the hot shower anyway, because I like it.