And if so, were they aware of their connection to the people there?
As far as we know, no, Tasmanian Aboriginals have been isolated from the mainland ever since the flooding of the Bassian Plain 8000 years ago, although its possible they might have ancestral memories of arriving from another land, in terms of actual contact there is no evidence for any contact. Also, the Tasmanian Aboriginals did not have any of the very useful technologies that the Australian Aboriginals had. So, again, its possible there were memories, but there was definitely no contact.