Sorry to say, I can confirm your aunt's sister is grossly exaggerating the extent of the warm season here, lol.
As the last 2 winters prove, Dallas in particular can get pretty cold at times in the Winter with snow/ice events. It doesn't stick around long and it's not always of consequences, but it does happen. Also, most of the trees do in fact change colors by mid-November before going bare like up north from December to March.
The September - November and March - May are actually pretty pleasant like a Summer up there, as it's not too humid yet and temps are only in the 70s and 80s.