After sending bitcoin to my Evercoin wallet through Coinbase, I backed up my account to an email address. Twenty or so minutes later my BTC was transferred without my authority to some random unknown public address.
This is wrong. You should never back up your private keys online, much less trust an email service to keep it safe. They are meant to be kept in an airgapped device or written on a piece of paper where it would not be exposed to hacks.
Checking through, I NEVER have copied or shared my private key, and checking my email security it is clear no one else has accessed it besides my devices and IP address.
I have no idea where the BTC went but got a notification on my phone at the same time that I received the amount in my account that was sent to that random address. But obviously, the BTC is nowhere to be found in the applications wallet.
Have you checked the transaction history on a block explorer, like
https://blockchair.com/ ? I have never heard of this 'evercoin wallet', could be a vulnerability from that end or you're using a corrupted file. Much better to use a reputable wallet like electrum and verifying the keys.
Does anyone have an idea where I went wrong here? If I accidentally clicked the recovery link in my email or did something else weird would my account send the BTC somewhere to keep it safe?
You did a couple of things wrong as I've already mentioned; storing your funds in a relatively unknown wallet and using email as a back up.
To the second question, I do not think there is any possibility of a wallet taking actions for the user like, locking accounts or sending out funds to
keep them safe. Except you are using a non custodian wallet, which this appears to be.
Or is it possible that my account was hacked and if so how could it have happened if not through email or me sharing a private key?
There are different possible scenarios;
• The wallet was compromised and hackers already had access to it and sent the Bitcoins out as they were received,
• There's also the possibility that the email was hacked and your private keys leaked in the process (I assume this is less likely as it would take a bit of time for the breach, except of course your email was already compromised).