What should I do for my under-eye texture and lines? (photos)

For texture concerns you can consider lasers, chemical peels, and microneedling.  PRP is another option.  These procedures are all done in an office setting.  It is also recommended that you use medical grade skin care products and wear sunscreen every day.

Under eye lines and texture can be permanently corrected with surgery or temporarily camouflaged with fillers (Restylane, Belotero, Volbella) which can also help nasolabial folds.There are great non-surgical options to fill in under eye depressions and hollowness. Using dermal fillers, a liquid eye-lift can be performed. A dermal filler such as Restylane would be placed in the cheek and tear trough to hide the depression or bag underneath your eye. 
Surgery would require a fat-repositioning lower lid blepharoplasty. In this surgical procedure, fat from the eye area is moved into the tear trough depression filling the actual depression. 
I hope this helps!