Getting around facebook messaging limit?

Hi,

I know that if I send too many private messages facebook will
tell me to slow down and then eventually block my ability to
send PM's... but if I simply send it from multiple accounts (send
from one, as soon as I hit the limit login to another) will that
get around it? Or are they looking at my IP? If so, is there a way
to hide this so that I can just keep switching from one account
to the next with no limit to how many messages I can send
from one computer?

I'm trying to send messages to people on fan pages
within my niche to offer a free giveaway that then promotes a paid
product. I look for fan pages in my niche, then look at recent
posts that have a lot of likes, then check the list of people who
liked it and open their page and send a message.

Thanks,

Bobby