if you already have a state ID or DL, then there is no need for you to carry your EAD with you at any time. only when you are offered and accept a job offer, and doing the initial paperwork for the job will you need to show your EAD.
there have been too many people who have lost their EADs because they carry it around with them. one SHOULD NEVER, EVER carry their EAD and SSN with them. if they are lost or stolen you will need to pay the $380 fee again in order to get another one.
also, if you do lose your SSN, you need to put a hold on it with the SSA in order to assure that no one else can use it.