You should be able to request a new IP if your plan includes a static IP. I have done this before for a business client, I personally don't think it was warranted but I just did what my boss told me to do.
If your plan does not have a static IP (i.e. IP is dynamic) then it is exactly that, dynamic, and it can change.
You have not said what type of internet connection you have, here in Australia I mostly work with clients who have ADSL. On just about every modem I have used, there is a setting on the modem's interface to specifically choose what your WAN IP Address is going to be. Default setting is AUTO. So when I change the AUTO address to the actual IP address that is statically
set, it still works. If I set it to a different IP Address, it actually stopped working.
So my point is if you have this type of setup (and a dynamic IP Address), then you could try manually "guessing" another IP address in the range of your ISP's allocated IP Addresses. If you hit one that is not in use then hopefully it will work? Sounds like fun to try, I would be interested to see if it works. I have a mailserver at home myself (hence static IP) so I can't test it out myself.
Otherwise if your ISP plan is static and they don't change it then that is poor service. I would change ISP and never look back.
However, two other comments come to mind.
1) ipconfig /flushdns from the command prompt actually does not change your internal IP address. You might be thinking of /release and /renew, which half the time it gives you the same one back anyway.
2) What on earth are you doing in the first place to attract unwanted DDoS attention?