If I may offer a suggestion - if you want to really see what disaster relief is like there is a way to 'try it on for size' before committing your life to it. I say this as a retired Texas National Guardsman and a certified FEMA level 800 disaster responder who helped run the J7 Office that was responsible for running the training and coordination for the TXNG's disaster response.
Why not try joining the TX State Guard? It's a small group of volunteers, mostly health care professionals, who meet every month to train and prepare for hurricanes, fires (basically everything under the Sun and Moon) to augment the in-state disaster response. You're not actually joining the military - you're joining an organization of civilians who 'serve' under the coordination of the Texas Adjutant General - the civilian arm if you will.
Why do I suggest this? Because the grass sure as fuck ain't green on the other side here - FEMA and state run disaster response mechanisms are always a cluster fuck at best, and I can say that from years of experience. There's never enough money/resources, everything dealing with government run events are run by job-preserving bureaucrats, the media is doing everything they can to either get in your way or indict you for incompetence (Anderson fucking Cooper should be tried for treason in my opinion - again, coming from personal experience) and it just goes on and on.
What you're talking about doing, leaving the job you know for one you're thinking is a dream job - hard but fulfilling, is a pipe dream in my honest opinion. Entitled shit bird, back stabbing parents will be replaced by entitled shit bird, back stabbing bureaucrats and strap-hangers.
Will you be able to savor the gratifying warmth of genuinely helping people in dire need? You sure as shit will, and it will uplift your soul - in my case it was a little old lady in Eagle Pass who could barely walk and stood up to give me a hug with tears in her eyes, I'll never forget that one. But conversely - have you ever really reached a kid in one of your classes and got them past their challenge and felt the relief and appreciation from him/her? My brother has (he's a teacher in Killeen for special needs kids), and when he talks about it - the grin on his face makes it obvious it's worth it to him.
tl;dr - Try it on for size first, join the Texas State Guard to augment disaster response in state.