If you don't live near the border nor are going to be traveling near the border, there's no reason to be carrying it around with you, you're just risking losing it.. You shouldn't be carrying your social security card with you either. Take a picture of your EAD and your SSN card if you don't know the number yet.
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Self filed AOS │Apps Received By USCIS - 3/18/19
│Biometrics Done - 4/11/19
Interview Scheduled - 4/24/19│Interview Date - 5/31/19│AOS Approval - 5/31/19
Permanent Resident Card Received - 6/8/19