The bridge of skin between the nostrils at the base of the nose is the columella. It can be elevated/reduced from a hanging position through either an open or closed rhinoplasty approach. The latter is used when columellar reduction is needed with few other modifications to the nose. Be careful not to just focus on one aspect of the nose; make sure you review with your surgeon a thorough analysis of the nose and what options exist for your individualized rhinoplasty. Worsening with smile may indicate the need for better nasal tip support. Nostril reduction (a.k.a. alar base reduction) can be performed through either the open or closed approach. Good luck. Keep educating yourself.
Your self asssessment is on the mark. Columella adjustment is quite a simple procedure and it will benefit you exactly the way you have described. Look for a competent plastic surgeon in your area.
When patients have a hanging columella, a columella-plasty is performed to reduce the amount of Columellar show. Excess skin and cartilage is removed from both sides of the columella either as a stand-alone procedure, or in combination with a full rhinoplasty. For many before and after columella- plasty results, please see the link and the video below