The first page just had a blank line of code on line 4. Without warnings turned on in your php.ini file you won't be able to tell why it isn't redirecting, but a page can't redirect after it starts writing content, and even a space or blank line counts as content. Removing the blank line allowed it to continue to the next page.
The thank you page still had old recordset code and wasn't entirely converted to MySQLi. I converted the remaining elements and that page displays properly now as well.