Using cookies with lynx
The following document implies that you can get this behavior by compiling Lynx with the --enable-persistent-cookies flag. The behavior is disabled by default, but implied to be enabled with this configuration flag.
You can use checkinstall or similar to easily create a package for your system from source.
Source: http://lynx.isc.org/release/lynx2-8-7/lynx_help/Lynx_users_guide.html#Cookies
Try setting -cookie_save_file to the same file.