Does applying for FAFSA make a school less likely to offer you funding? I have applied to a number of Ph.D. programs in mathematics, keeping in mind up until this point that the generally accepted wisdom is: "If you aren't fully funded, don't go."

Now, luckily, I have received full funding (TA position and full tuition waiver) from one university, and I still have five more places I'm waiting to hear back from, but the one funding offer I've gotten so far is from one of my safety schools. 

I got accepted into another university that I would much better prefer attending, but without much of a chance of a full funding offer. 

Now, I have already taken out significant (~40,000$) student loans for my undergraduate degree, and by no means plan on significantly increasing that to get my Ph.D, but I came to thinking: "What if I got accepted into a really good school, got decent, but not 100% funding, and decided to take out a small loan to attend?" and so I decided I should fill out FAFSA "just in case".

However, now that I am filling out the form, given that it requires you give a list of universities to send your information to before you receive information on how much funding you are eligible for, I wanted to know whether or not supplying this information to the schools I haven't heard back from decreases my chances of getting a tuition waiver or TA position from them, since they see that I already have funding? (which of course I don't want)

So is this a legitimate concern? Or should I go ahead and fill out the FAFSA?
You should not worry about this. First of all you have already applied and usually you are not supposed to add any additional documents to your application. Secondly these information are separate so I doubt one will influence the other. And I wouldn't want to go to a program where they make you pay more to save money.