Creative Cloud App continues to tell me that I need to purchase Lightroom but I already have membership.
One possibility is that you have signed in with the wrong user ID. You need to sign in with the exact same user ID that you used when you purchased the subscription.
And don't shout at us, we are not Adobe here, we are other LR users who don't know anything about your subscription and can't do anything about it.
If that's not it, perhaps there is some troubleshooting that other users can chime in with, but if not I'd bet that Adobe Customer Service (phone or chat) could address this quickly - go to Adobe.com and follow the Support > Contact Adobe links.