Yes, submit your change of address immediately. Although it may not have any consequences, technically you have violated immigration regulations. But have no fear, they will probably just look the other way.
There are almost always issues for appeal, especially in a murder trial, so I would strongly suggest you contact an appellate attorney. They are not cheap though, but I think it's worth the effort. Otherwise, depending on priors and other evidence, you may be looking at 15 years behind bars.
I believe you're omitting crucial information. Did she endorse the check, did she know about your claim? Answering these questions will be important to determine whether she is entitled to a portion of your settlement, which I believe is what you want to know.
More information is needed, but usually with a serious injury such as yours, if you can prove negligence and causation, then there is a possibility for recovery. Contact a personal injury attorney for consultation.