The problem you are describing is usually (but not always) related to strong muscles at the base of the nose which pull the nose down during smiling.
If this is the only reason your tip is pulled down, the muscle can be divided under local anesthesia with oral Valium or similar medication.
It's probably okay that your cast came off a day or two early. I would let your surgeon know just to be sure, though.
The cast is mainly there as protection (like wearing a helmet for your nose).
The versatility and safety of Gore Tex nasal implants has been well described. The likelihood of extrusion is less than 5%. If you're pleased by the appearance of your nose, enjoy your new look. I hope this helps you.
