Hi Katheek 77!
I think your experience will be looked on with favor. Assuming you are eligible, like you mentioned with the 5 year window from college, and Fulbright's commitment to a diverse representative body - I can't think of any logistical reason you'd be ineligible. Being that you are mid-career professional with experience and an advanced degree, I think the review committee's central question when looking at your application is "why now?" It will be important to articulate in your application the next steps after Fulbright, despite having the master's program clinic hours (which I'd confirm your location is willing to sign off on before acceptance) and presumably a job after ESL degree, you must clarify how being a Fulbright Fellow will better prepare you to be an agent of positive change in cultural exchange, language learning, and professional development. I've read about successful applicants ranging from 25-50, it may be an asset to stress the new-ness you mentioned to add some vitality and curiosity to your app!
Also, it goes without saying you must apply to a location which accepts Fellows with dependents. What a fun experience for an 11 year old!
Best of luck