When creating a booklet the resulting PDF documents shows headers just before page break, even though I have added the "page-break-after: avoid" option in my print style sheets.

h1, h2, h3, h4, h5
{
  font-family: Trebuchet MS, Verdana, Arial;
  font-weight: bold;
  page-break-after: avoid;
}

How can I keep this from happening?

You can choose to have a page break before each topic in the booklet properties. For specific headings falling in a place you would rather avoid, you could put a specific page break style on the heading element e.g. - style="page-break-before: always"