You should fill out the form but you should not be surprised if it is denied.

The point of the form is generally not for people in your situation because Rutgers is well within their rights to charge you out-of-state tuition; your wife literally won't be living in NJ so why should she get in-state? The form is mostly for people who do live in NJ but due to a technicality would otherwise not receive in-state. For example, by default a dependent student receives in-state tuition only if their parents also live in NJ; however there are many dependent students who do not live with their parents and it would not be fair for a dependent student who lives in NJ to be charged out-of-state just because their parents aren't in NJ. Again, the fact that you're actually living in NJ is the most important part.