Question about cellphone and Sim card

Few day ago accidentaly droped my non smart cellphone in a poddle of water and although the phone is working fine, the battery will not hold charge.
Phone is GSM type with AT&T newtwork.

So just going to get a new phone.

My question is if I exchange sim card with a new [locked] cellphone that also uses ATT newtwork, would that work or do I need a unlocked phone?

Thanks.