I am a new shia convert in a strict sunni family. In some occasions, I pray with my family or around them or even in a sunni mosque. Are my prayers accepted if I pray with my hands crossed and without a turbah (or anything natural under my forehead) or should i re-do my prayers once I am alone?

In your circumstances your namaz is acceptable if praying in Sunni mosque or in front of your family .
If it is possible please pray when no one is there so you can pray freely.