Today, I was eating with my family. I had a rough idea about the suhoor time, but wasn't sure. I continued to eat but realized it must be time so turned on the TV, and realized it has passed. I had stopped eating a couple minutes prior, so I'm not sure if I stopped before the time or after. Can I continue with fast today?
It seems at your location you can't hear the adhan.
So basically you might have unintentionally continued eating while you should have stopped. So far if there was any mistake or breaking of fast it was unintentional and therefore it may hardly count as such. To clear this you may consult a scholar or mufti of your trust. Some may say it is OK others may say you'd better expiate by fasting one day after Ramadan.
If you ate afterwards you are certainly and intentionally breaking your fast which certainly is sinful and should never be an option as the expiation for that according some schools of fiqh might be much harder.