After you've done everything at the link above and have the new iCloud account up and running, you can still add the old iCloud account back in "Settings > Accounts & Passwords > Add Account". This way you can easily access the old notes and old emails that don't get migrated over when you sign into the new iCloud account.
Assuming that you allowed the old contacts, calendars, reminders, safari, passwords, etc to stay on your iPhone as you turned the sync off one by one, before you signed into the new iCloud account, then those would have been migrated over to the new one.
Doing the above would just add the old iCloud account as a standard internet account, just like any gmail, hotmail, or yahoo account you might have added. But once added into the accounts section of settings (instead of the iCloud section of settings like the new one), you can enable just the old mail and old notes there. You wont enable the old contact, calendars, and reminders though, as they should have moved to the new iCloud account. When you are in notes or email app, you can choose which account you want to view, the old one or the new one.
You can also keep using the old iCloud account in the App and Music store to access your old purchases, and I do this so that I don't have purchases in both iCloud accounts - I use the new one only for syncing private data, and the old one for any purchases.
But if you sign into the old iCloud account as above, you could at least avoid having to move the notes. That just leaves photos, and if you have enough storage on the iPhone then it's not an issue to download all the photos first.
It's a shame there isn't an easier way to migrate your photos from one iCloud photo library to the other, but if you have it set to "optimize iPhone storage" then you have to change the settings to "Download and Keep Originals" and then wait hours for it to complete the download process before you can turn off iCloud photo library - you may not have enough room or bandwidth on your device if the library is huge.
It's a non-issue if you never turned on optimize, or never turned on iCloud photo library.
