[Cryogenica]

Yes sadly my ISP doesn't allow for that much customisation with my modem.

It wouldn't be your modem you customize, it would be your router. Nevertheless, some modems do double as routers. Perhaps that's the case for you?
If you do have a separate router, you should install DD-WRT (or some WRT variant), assuming it's compatible, or if it's not compatible, I'd recommend getting a router that is.
But, of course, if your modem and router are the same unit, I have to agree with other comments that you're pretty much screwed until you get a new router.

But there is another way. Usually the ISP reserves your IP for either 24, 48, or 72 hours. If you can go with internet for that long (I'd add at least an hour padding, as well) you should be able to get a new IP address. If that doesn't work, you probably have a static IP so you would need to either get a new router, or if that doesn't work you might even have to get a new internet plan.

My modem/router are indeed 1 unit and I unplugged my entire internet for a whole week and still no new IP, guess I'm screwed.

Then your last chance would probably be to buy a new one or get a new internet plan, depending on your ISP. Or you could see if the attacks stop coming eventually, but that might not happen for months to years.