How Can I Refresh the Appearance of my Undereye Area?

Your primary problem is volume deficit in the lower lids.  You can't reproduce the finger maneuver with surgery.  I'd suggest you start with Restylane done by a doctor, not a nurse of aesthetician to plump up the hollows.  This can last a year or more.

In the photos I see significant hollowing around the eyes with what appears to edema on the left just above the tear trough. If you have any nasal allergies those have to be addressed and controlled first otherwise any surgery or procedure you get will only have short lived results. If any present allergies are controlled alot of this swelling and apparent skin excess will go away. If that is not the case the most reliable way is to get rid of the excess skin by surgery. You will still... more