Activation

The modern equivalent to Lightroom 6 which is no longer supported is Creative Cloud Lightroom Classic.

Even if you manage to install and activate on your new laptop, there will be no more updates for it. 

Consider getting a Creative Cloud Photography Plan (approx $10/month for 1 year subscription). The latest software is compatible with new computers and cameras.
https://www.adobe.com/creativecloud/photography.html

As of Adobe policy, you can activate the software on 2 computers.

@linsims: OP say Photoshop LIGHTROOM 6. The Photoshop comes only from the very bad idea form Adobe the call everything that is a pixel editor "Photoshop"...

And yes, Linsims is correct in the sense directing you to Customer Care. Adobe customer care can be contacted here: https://helpx.adobe.com/contact.html - click the small callout at the lower right of the screen and type "agent".

This is a public forum, not Adobe support. Users, including the ACPs and MVPs, can't provide support for account, subscription, billing, download, or product activation issues. Support for those issues requires an Adobe employee. For Creative Suites (if that's what you have given you mention that it is Photoshop 6), the only help they're able to provide is resetting your activations. Luckily, that's what you need done.

Click the following to contact Adobe customer support staff for help: 

https://helpx.adobe.com/contact/support.html

You must be signed in with your Adobe ID and allow cookies in your web browser for this to work.

This is the link to the US website. If you are not in the US, the site should redirect to the page appropriate to the region associated with your ID.

Click the chat icon at the bottom right of the page to open a chat session. Typing AGENT into the chat window should connect you directly to a CSR rather than the AI.

If the CSR refers you to the forums, insist on speaking with a supervisor and keep moving up the chain until you get someone who knows what they are talking about.

You can also search for Adobe customer support using your favorite search engine. This will normally provide you with a regionally appropriate phone number that you can use.

Be aware that because of the pandemic, wait times are far longer than usual due to reduced staff and higher demand. Please be patient with the people to whom you speak. Like you, they are feeling a great deal of stress. A little patience and understanding will go a long way to helping you in your particular situation.