If you are moving to a new address, you do not need to buy new renters insurance. Instead, you can just transfer it to your new address by making a phone call or visiting the insurance carrier’s website to fill out an application form.
The information in this answer is general in nature and is not legal or insurance advice for your specific situation. Policy language varies carrier by carrier; your declarations page is the final word. If you're unsure, call your agent before you need to file.