Ack! We have the same problem...
Except we're a private school. Which can be just as complicated.
What we have to do is run our letters through our Communications people, just to check if any of the people we're trying to get to sponsor us already sponsor the school or something like that.
For instance: based out of Atlanta, an obvious sponsor would be Coca-Cola, especially since many parents of kids at the school work at Coke (including my mom. And the CEO and several board members had kids at our school). However, we're not really allowed to ask them since our school has a partnership with Coke, so that didn't work out. (YET
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Even though there are obvious differences between private and public schools, there should be someone who has a list of potential sponsors that have been sought in the past. Now the question is, is that person at your school or part of the larger school district system?
For a time, we were banned from approaching anyone to sponsor us. Now we're given a very limited degree, but still better than before, and running letters by the Communications Director cleared things up for us.
Best of luck!