I've been able to get new IPs by simply power cycling my IP - though my IP also used to change randomly on its own also (I always presumed there was some network outage in the night when this happened though), I've also had quite a few different ISPs and they all worked like this, maybe it's more common in the UK.
I only know this is the norm because I host a website and media server from my home and had to update my DNS records everytime my IP changed, so I started to request static IPs
I'm not sure where you're from, I don't know if it's common where you are, but as far as I know here in the UK there is only 1 ISP that offers static IPs as standard. Otherwise it is something you have to request from your ISP
Easiest way for most people to get around an IP ban is to simply turn your router off for a few minutes - most ISPs will recycle that IP and assign a new one to you.
Doesn't work if you have a static IP, but that is usually something you have to ask for specifically
Any chance it only redirects the watch URLs on YouTube? So you can use YouTube to find the videos and on clicking them links it takes you elsewhere?