So I've been getting EA verification codes texted to my phone almost every day now from what I can only assume is someone trying to access my account. Now I'm getting emails that I can't tell if they are phishing or not, but they say my EA account is being accessed through the web.

Long question short, do I have anything to worry about? There haven't been any games played on my account since I last played, thankfully. I've had 2FA enabled for years, just changed all my passwords (again) but someone clearly has it out for me and I am worried about my account being compromised / rage hack / ban. 

Maybe try to reach out to EA by yourself. They'll most likely give you all the options you have to secure more efficiently your account and perhaps try to identify who's looking to access to your account.

Also check if your mail (the one with which you made your EA account) hasn't been hacked. I suggest to change your pw there too and enable 2FA as well.

Change your email