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Re iCloud being full, you need to buy more iCloud space. Follow the steps below to buy more storage.
Go to Settings > iCloud > Storage. Tap Buy More Storage or Change Storage Plan. Choose a plan, then tap Buy and enter your Apple ID password. This will upgrade the plan and thus allow you to store more contents
(Also, If iCloud library is enabled then go to Settings > iCloud > Photos and enable "Optimize Device Storage". This will save phone storage)
Re your email issue, what happens when you access your email ?