Hi Eli, welcome to the PCHF.
IP addresses are normally given out by the ISP as "dynamic" which means each time you connect to the ISP...you will get a new one. Cable is always connected so it's very rare to get a new IP address - the fact you've just got a new router explains this part.
You can request to have a "static" ip address which is yours until you say otherwise - and it costs a bit more normally...Unless you're hosting a web/mail server their is no need for a static ip address.
Hope this helps and if you need anything else, just ask away
