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Akeeba Backup is a full site backup and restoration solution. "Full site" means the entirety of your site's database content and files. The idea is to restore a site to its previous state OR create a pristine copy of a site on a different location.

While you can make a partial copy of your database, in your case it is NOT recommended. Articles and categories are interlinked. You could argue that since you're transferring both this is not an issue and you'd be correct. However, they also have links to the #__assets table of your site which controls access control for articles, categories, extensions and everything else. Therefore your articles and categories are effectively interlinked with EVERYTHING on your site, including every single menu item, module, plugin and component. That's what prevents a partial transfer of just articles and categories and that's why we have abandoned that idea since Joomla! 1.6 back in 2010.

Instead, the workflow with Akeeba Backup would be like this. Take a backup. Restore it somewhere else, e.g. your own computer. Remove unnecessary extensions. Upgrade Joomla. Install the new extensions. Design the site around it. Take a backup. Upload to live.

If you have already designed the new site you need something like SP Transfer. However, please note that doing this you will end up using different article IDs in the new site, therefore your links -which always include the article ID- will be broken.

I don't know which of your two options is better: redesign the site or figure out how to write loads of redirections from the old site's URLs to the new site's URLs. This is up to you to decide.