Quite right Backwoodsman. Voiced and unvoiced dental fricative.

I'd say 'think' would indeed be 'fink', But 'that' might sometimes have nothing at all at the front, as in 'wossat?' (What's that?) or 'oozat?' (who's that?)
My sis and I used to say 'aas' for house. Drove my poor father mad.

To complicate matters, my mother (from Cork) spoke with a soft Irish accent, and used 't' for 'th', as in Tursday,
It's a wonder we learned any English at all!