I have not tried to do this, but have been tempted. From what I have heard, some guidelines:
- When you call, be polite and professional. Stay calm at all times. Be aware that the best that is likely to happen is that whomever you talk to will listen, note what you say, and state that they will consider this. It is unlikely that whomever you get on the phone actually has the authority to up your offer. Most likely whomever you talk to on the phone will then send your information to the person who gets to decide.
- The total cost of attendance (for the full four years) is probably more important than the size of the scholarship. You should calculate this carefully before you call.
- You message above seems to be "I prefer Tulane, it is a great university for me, but it is more expensive than the alternative". I think that this is a very good general message.
- I am not sure where to call. However, look at your offer letter and see if there is an obvious number to call. Alternately, call the admissions office at Tulane. When you get someone explain why you are calling, and ask whether they are the right person to talk to.
- It might not hurt to have a few good reasons why you would prefer Tulane in mind before you call.
- If you come across as a serious student who has good reasons to prefer Tulane, and who is likely to do well there, this may help you chances.
- It is probably better for you to call rather than your parents. You are the person who will be attending, and you are the person who really likes Tulane. People at Tulane should be able to understand that your parents don't want to pay more.
One question: Are there other schools that you are waiting on that could change the situation? If not, then I would be inclined to go ahead and call. The worst that is likely to happen is nothing. I think that this is worth the try.