When a cookie is set for a domain, the browser sends it for every request, as long as the cookie is not expired. So no, there is no way to say "don't send cookies for this request" except a different domain which you mentioned.
Depending on your website, you can, instead of using cookies, use:
- query parameters: GET my_file.php?my_session_id=abcd123456
- custom headers (for ajax requests only): you can set HTTP headers
X-MY-SESSION-ID: abcd123456
- POST body variables
You asked for alternatives, so I gave some but it's hard to implement these if all you want is to avoid sending cookies when loading a static asset.
So my suggestion is to look at how many extra bytes you're actually sending for each request and trying to reduce that to a minimum.
In my opinion, only one cookie is really needed for the user's session id, so that's about 50-60 bytes. You may also choose to use Google Analytics, which will add about a couple hundred bytes. Everything besides that is bloat. But don't sweat it if Yslow shames you for 300 extra bytes, because it won't make a huge difference.