I need to run my LR classic on more than 2 computers

Purchase a license of Adobe applications on two Adobe IDs of yours. Then you will have activation on 4 license and collaborate your cloud files by using sharing through email with Adobe Cloud.

Share Adobe Creative Cloud files, libraries, prototypes, and more

You can install Adobe Creative cloud desktop applications on your 3 devices and Whenever you need just sign in Adobe creative cloud desktop application and it will activate your license on all the Adobe applications you have.

How to download Creative Cloud desktop app

Or the best way is

You can purchase the team license of Adobe Photography plan and easily access on more than 2 computer

https://prodesigntools.com/creative-cloud-for-teams-differences.html

It’s not possible to run Lr on more than 2 computers, without deactivating.

I would recommend to ask Contact Customer Care .

I would however wait until after this week.

You need to either get a team license, or buy separate retail subscriptions under different Adobe IDs. Which means you probably can't share your stuff easily. Not ideal...