I am assuming that Family Care is a Medicaid type program. There are limits to public benefits that can be obtained by new GC holders; since you do not yet have a GC, it is unlikely that you can receive any and may be denied the GC by becoming a public charge. If you were sponsored by a family member, they should have the resources to support you; if they do not, the state can recover the money spent on your behalf from them.
You need to check to see which benefits you are qualified to receive in your state both as a new GC holder and as a foreign national (which is all you are at this time).
