OK, I found the problem, although I don't understand all. I was using Process.Start(); but then I saw an example using Help.ShowHelp(); This is the event handler for the dropdown help menu (not F1 key), so I don't know why it fixed the F1 key case as well, but it did. The multiple behavior is gone.
private void menuItem6_Click(object sender, System.EventArgs e)
{
Help.ShowHelp(this, "Flash_Help.chm");
}