Even if your new home state offers trailing-spouse unemployment benefits, you may not be entitled for them if you do not meet other rules. As your job is non-transferrable, so in order to protect yourself and ensure your future benefits, you must try to give your current employer at least 30 days notice before leaving the job.
In case you are forced to move on shorter notice, document this in order to support your claim in new state.
File for unemployment benefits in your new state as soon as possible to ensure you miss no deadlines.