Home IP rarely change unless you have a dynamic one, even then it takes a while for it to change while DCed or restarting router.
A new router and PC doesn't change your IP, the IP is on the internet providers side.
Obviously, there has no "botting safely" after getting a ban or warning for botting. You probably had the same IP. so chances are botting on your own home IP will result with more bans. Starting a new account means nothing unless you make/bot it on a proxy. Even back in the day, this was a given so it should be clear things are harder than before.
You can check if your internet provider provides a static or dynamic IP by going to the router/modems settings through the default private IP. Google your model number to find the private address and the default login (unless you've changed it before then you should know how to access it). Once you confirm its dynamic, you can edit the lease settings (rare nowadays from what Ive seen) to reduce the lease from XX minutes to say 5 min. Then save, turn off your router/modem, disconnect all cables to make sure, then simply wait more than 5 min and reconnect it and reset the lease settings back to normal.
If youre lucky simply releasing/renewing the lease will DC your internet for a sec and reconnect with a new IP.
Otherwise, dont bot at all on your home IP and just get a cheap proxy that hasnt been blacklisted by Jagex. You could always risk botting and mixing a lot of play by hand but I wouldnt