You'll need to figure out what sort of cookies your sites are actually using, what domains they're from, and verify if they're being saved or not.
If not set your exceptions properly.
I recommend you use the Chromium internal pages to set exceptions so you can use wildcards properly.
Site Data (first-party): chrome://settings/content/siteData
Third-party cookies: chrome://settings/cookies
This is easier if you're used to Chrome settings.
Note that the third-party cookie settings page is kind of broken sometimes, it might show up as blank. If so just go to chrome://settings/ and into Third-party cookies from there.
Note also the wording on the third-party cookies page:
Sites allowed to use third-party cookies.
This was changed several Chromium versions ago, it used to be:
Sites that can always use cookies
Because at the time that page controlled all cookies, not just third-party.
This basically means that if you add [*.]bbc.com to "Sites allowed to use third-party cookies" it will allow all third-party cookies on BBC.
It used to be you could specify explicit exception what sites you wanted to allow third-party cookies from.