I agree with the above that it’s going to be tough to get money at this point in the game, especially for an OOS public, with no FAFSA completed. Merit awards are not typically available for transfer students. I think if you are instate for Rutgers, that’s your best bet for seeing some $.
I also think that a lot of campuses are looking to de-densify campus so are not looking to have everyone they admitted accept. I believe OSU is no longer guaranteeing housing for transfers.
Purdue had a record yield this year and spent $50 million in covid preparations for the upcoming year so I very much doubt they would be willing to “negotiate” either. This is from Purdue’s FAQ on their financial aid website: "Purdue is unable to match financial aid offers from other schools and does not negotiate aid eligibility. " (My D is a rising junior at Purdue so they’re the school on your list I’m most familiar with).
I think you need to prepare yourself for your child to be full pay but give it a go.
Please circle back and let us know how it went. Good luck.