Hey Toka4ever,
We understand that you're wondering if you have to use iCloud or if you can use another service, along with that your iCloud is full. It depends on your preferences if you'd like to use iCloud or the storage service from another provider, like your mobile carrier.
Messages can be saved in iCloud or computer backups: About backups for iPhone, iPad, and iPod touch If you have Messages for iCloud enabled, then those are stored in iCloud and will appear on any device that you have Messages for iCloud enabled. Those cannot be saved elsewhere, but if needed, you could take screenshots of your messages that you do not want to lose: Take a screenshot on your iPhone That would save the screenshot on your iPhone, which you can keep there or you can transfer that to a computer: Transfer photos and videos from your iPhone, iPad, or iPod touch You can also transfer your other photos to a computer as well.
You can see how to archive other data in iCloud: Archive or make copies of the information you store in iCloud If you'd like, the steps below from this article show how to make more space in iCloud: Manage your iCloud storage
If you run out of iCloud storage, your device won’t back up to iCloud, new photos and videos won't upload to iCloud Photos, and iCloud Drive, other iCloud apps, and text messages won't stay up to date across your devices. And you can't send or receive emails with your iCloud email address.
You can free up storage in iCloud by deleting content that you're not using:
Before you delete anything, you might want to archive or make copies of the information you store in iCloud."
You can also read more about photo storage through AT&T here: AT&T Photo Storage
We hope that information helps you. Best wishes!