Hey MayaLintu
If the phone is so full it can't do anything you may want to try a few options.
Sync your iPhone, iPad, or iPod touch with iTunes on your computer using USB - Apple Support. This will help connect to your new "main computer" although this will delete any synced data.
Also keep in mind that backing up your phone only backs up certain features, iCloud and iTunes have different backup criteria.
About backups for iOS devices - Apple Support
What does iCloud back up? - Apple Support
When i was in this situation, I deleted as much as I could including Apps and then ran a backup and also backed up my photos and videos to iCloud.
When you sync your phone to a new computer you will keep your apps, just if you sync music then you will have to import your entire library back over.. I can't stress enough this is not included in your backup.
But you can redownlaod them from the cloud, if they are purchased through iTunes.
Redownload music, movies, TV shows, apps, and books from the iTunes Store, iBooks Store, and App Store - Apple Support
Hope this helps.
