Looks like Rutgers is one of your better choices, since it does not appear that there are any special barriers to declaring or changing to an engineering major there (ask directly if you are not sure): http://soe.rutgers.edu/oas/first-year-declaration
In comparison, Purdue engineering majors are competitive-entry for first year pre-engineering students: https://engineering.purdue.edu/ENE/Academics/FirstYear/T2M
At UIUC, it may be difficult to change to majors which are “full”, which some engineering majors are: https://engineering.illinois.edu/academics/undergraduate/advising/changing-majors.html
Maryland also does not appear to have any special barriers to declaring or changing engineering major (ask directly if you are not sure): https://eng.umd.edu/advising-faqs
Not sure why you would want to pay more to get weeded out at Purdue, or locked into whatever major you were admitted to at UIUC.
Also, if your EFC is only $12k, but your cheapest option is $35k, can your parents afford more than the EFC? If not, it may be that all of your choices are too expensive (you cannot take that much in loans without a cosigner).