Totally agree. Also, Splinterwaifs wouldn't exist if the dryad died completely when the tree was killed.
So, if the DM has already ruled that a dryad and her tree are literally one being, then they have more sameness than a pair of dvati twins. Teleporting would be an issue, first because a dryad would count as a much larger creature for the spell, one medium creature and one huge or larger creature (or maybe just one super large creature) because their trees have to be so big. The other issue would be that wherever you went, a giant tree would show up. If you teleport into a space that would be fine for a medium creature but unsafely small for a gargantuan oak tree (most indoor places), one or both parts of the dryad tree combo would suffer the scrambling mishap damage. That could be double damage, too!
Also, it's almost as if they have 'share spells' like with a familiar, only more so. I agree that under this ruling you would have flying trees, and if you used a shape changing spell the tree should disappear from wherever it is and merge with the dryad's body. But unlike 'share spells', this would also be the case with a non spell effect like Wild Shape. Plus, which one would merge into the other? You'd have to decide if you went by proximity to the originating effect, or by mass, or maybe they end up changed but somewhere in the middle as their bodies physically move and merge? What if something prevents them from moving? Do they act like an ooze or something with gaseous form?
Then, you'd have to decide if the tree was still an object, or, because it was part of a dryad, had some portion of the dryad's hit points and other vulnerabilities. That would relate to the mishap damage from teleport too...