Depends what kind of traveling you want to do, you couldnt fly with it, but just as an id in case you got asked for you, sure, that plus your expired passport.Ask a similar question
Valid one, you can. It is a Government issued ID, after all.
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Your EAD card is not a travel document but since is newly issued you should immediately apply at local DPS to renew your DL. You can use both or either as an ID or if you have a valid passport to fly within the U. S.Ask a similar question