What your asking sync will not do or do or miss something you need to do more complete data transfer if you want your profile saved correctly. Once your old is gone any sync will be gone along with it.

https://support.mozilla.org/en-US/kb/back-and-restore-information-firefox-profiles https://support.mozilla.org/en-US/kb/restore-bookmarks-from-backup-or-move-them https://support.mozilla.org/en-US/questions/966133 https://support.mozilla.org/en-US/kb/export-login-data-firefox

Sync isn't a backup for important bookmark or logins if the new computer doesn't connect to sync.

If it doesn't work via Sync then you can copy logins.json and key4.db directly.

You can copy certain files with Firefox closed to the current profile folder to transfer or recover personal data. Note that best is to avoid copying a full profile folder.

You can use the button on the "Help -> More Troubleshooting Information" (about:support) page to go to the current Firefox profile folder or use the about:profiles page (Root directory).

• Help -> More Troubleshooting Information -> Profile Folder/Directory:
  Windows: Open Folder; Linux: Open Directory; Mac: Show in Finder
• https://support.mozilla.org/en-US/kb/profiles-where-firefox-stores-user-data

• bookmarks and history: places.sqlite
• favicons: favicons.sqlite
• bookmark backups: compressed .jsonlz4 JSON backups in the bookmarkbackups folder
• cookies.sqlite for the Cookies
• formhistory.sqlite for saved autocomplete Form Data
• logins.json (encrypted logins;32+) and key4.db (decryption key;58+) for Passwords saved in the Password Manager
  key3.db support ended in 73+; to use key3.db in 58-72, make sure to remove key4.db
• cert9.db (58+) for (intermediate) certificates stored in the Certificate Manager
• persdict.dat for words added to the spell checker dictionary
• permissions.sqlite for Permissions and possibly content-prefs.sqlite for other website specific data (Site Preferences)
• sessionstore.jsonlz4 for open tabs and pinned tabs (see also the sessionstore-backups folder)