Deborah Kate:
Without an in-person evaluation I can only speculate about what you may be seeing, and I do not know what was done during your surgery (there are a variety of techniques used during facelift / necklift).
Vertical bands caused by the underlying platysma muscle are one of the primary issues treated by a facelift / necklift operation. Botox alone is much less effective, but may offer a little benefit in addition to your surgery. Clarify with your surgeon if your procedure included steps designed to correct to the muscle bands.
Horizontal skin wrinkles may correspond to the rings you notice. These typically do not go completely away after necklift. The ones lower in the neck are most likely to remain. Surgery improves skin tone and underlying structures, but your skin cover remains. Additional treatments for skin texture may be helpful. These include cosmeceuticals, intense pulsed light, non-ablative skin tightening, or laser resurfacing. These are often done as a series of smaller treatments, followed by periodic maintenance visits. Seek a physician experienced with non-surgical and minimally-invasive skin treatments (usually a board-certified dermatologist or plastic surgeon).
