I can't get into purchased Lightroom.

Is it possible the OP subscribed to the Lightroom Plan and is trying to use Lightroom-Classic?

Lr-Classic requires a subscription to the Photography Plan.

You have to use the exact same e-mail address to sign into Lightroom Classic that you used when you purchased the subscription.

When do you have buyed the subscription? Sometimes it takes time until the subsription is valid.

In a first step sign out and sign in again using the Adobe Creative Cloud desktop application (Sign in and sign out to activate Creative Cloud apps)

If this doesn't help try the next step. It's to reset the preferences to the default settings.

• Reset Procedure:
• Close Lightroom.
• Windows: Hold down [Alt]+[Shift] while restarting Lightroom.
  Mac: Hold down [Opt]+[Shift] while restarting Lightroom.
• Overwrite the Preferences when prompted by the dialog.
• Close Lightroom.
• Restart Lightroom.

It's recommended to backup your preferences before you reset the preferences to the default settings: 

https://helpx.adobe.com/lightroom-classic/kb/preference-file-and-other-file-locations.html

If this doesn't help please try this: Purchased Creative Cloud apps appear as trial versions (adobe.com)