Cannot post new content after installing varnish
I finally found a solution. Inside my default.vcl file I added the following block at the top of sub vcl_recv:

sub vcl_recv {

  # forward request directly to Apache if SESSION cookie is set.
  if (req.http.Cookie ~ "(SESS[a-zA-Z0-9]*)=") {
    return (pass);
  }

  ... # whatever else you need to do in vcl_recv
}


Basically I'm just checking to see if my session cookie is set (anyone else reading this might have one formatted differently) and if so, forwarding it to Apache. This seems to have fixed the problems as new content can be posted again. Not really sure why the elaborate cookie filtering I had in place (and which was working in every other case except posting new node content) didn't work, but this seems to be the solution, even if it feels a little heavy handed.