Strange new lines and rough skin after Botox to DAO. Could this be caused by the Botox? Will they disappear?

While Botox and other neuromodulators have no direct effect on the skin, the muscles paralyzed by the effects of these medications do. There are many facial muscles in the area of the angle of mouth and any skeletal muscle may be affected by the paralysis caused by Botox. It is possible that since the DAO is paralyzed other muscles in the general area that would normally be counteracted by the DAO are exerting different tension vectors on the skin. Fortunately, the effects of neuromodulators are temporary and will definitely resolve between 3-4 months after injection. If you are unhappy with this appearance I would advise against any futher Botox injections in the perioral area.

It would be helpful to see pictures of your face at rest and smiling.  However, I am not sure I can imagine Botox being responsible for the wrinkles you described

Skin changes are unlikely related to your Botox. However, lines re-appearing in that area may be a sign that the Botox is wearing off. Visit your injector for their opinion.

It is not likely the cause as you surmise. Botox does not have any detrimental effects on the skin. Seek a skin specialist who can help you enhance the fine lines with fractional surfacing, Ultherapy, retinoic creams, etc 

There are other reasons for the change in skin texture and new lines.  It is not the Botox.  Perhaps a different skin care regime?  Other skin treatments?  Botox causes no permanent changes.  Botox effects will last about 3-4 months.