Struggling to distinguish between ة and ه in Levantine speech

I’m a heritage learner trying to sound less robotic when talking to my grandmother, but I keep tripping over the 'ta marbuta' versus 'ha' pronunciation. In my house, they seem to drop the 't' sound entirely, which makes it hard to distinguish between standard Arabic grammar and the local dialect. How can I train my ear to capture these subtle shifts without sounding like I'm reading from a textbook?