...And tell them what? "Oh, can you change my IP address because I keep getting banned on
RuneScape--you know, that MMORPG--and I would like it if I wasn't flagged any more."
You're probably being banned because you...
Are using multiple accounts to bot the same activity, and their system is already trained to detect and remove.
Are paying for membership using the same credit card/phone number/profile(able) method, and they're simply finding out who else you've paid membership for, monitoring them for a short time, and concluding whether or not they're botting.
Are creating new accounts from one IP, and then sending them off down another, not thinking for a minute that maybe, in this day and age, account created are also recorded with their creator's IP address. And of course, finding other accounts made from the same IP, checking them out, and removing.
Either of these methods will get you screwed, and I can't imagine them being too hard to implement for Jagex.
If you want a solution, well there isn't a straight forward answer. How about testing the waters; create a new account from your IP, bot it via using an obscure, but still functional, script, and see if you're flagged enough to get caught. And what I mean by "obscure" is, something that you're less likely to get reported for.
And also, I would think that free proxies only distribute Class B IP addresses, because they can offer more hosts (aka. customers) than network spaces. While I'm no pro in networking, I do know that standard networks that we have, use Class C IP addresses. So having an account switch between IP classes would be greatly suspicious. Why the hell would you trust a free proxy any ways?