You are signed in with the Adobe ID (e-mail) associated with your active plan, correct? Once you're on the "Contact Customer Care" page, go through the filters choosing Creative Cloud All Apps > Lightroom CC > Download and Install > Show me my contact options. Does it still show only the forum? If that's the case, there may be a bigger issue with respect to Adobe recognizing your subscription, and you should go back to the beginning of the filters and choose "Account Issues" instead. That should let you get to a chat agent.
Good luck with the resolution. Let us know how things progress.