I have been having a big problem with text moving after saving posts & pages.
Has anyone experienced this?
How can it be fixed?

Need more info, link to example etc.

I manually type in html code like <br> and <center>, switch back to visual mode and save and the hand coded html is gone.

That is the way WordPress works. A break tag is just a new line in the HTML editor and <center> is a depricated tag. You should be using text-align.