Any form of pure horizontal chin augmentation will make the labiomental fold deeper, regardless of whether it is an implant or a sliding genioplasty. Bringing the chin forward and down (vertically lengthening) will help diminish any significant labiomental fold deepening.
Cost factors vary significantly for a number of good reasons: Prices vary greatly because the surgery itself encompasses a wide range of techniques that can be performed to different extents. My recommendation is to first find the most qualified surgeons where you want the surgery ...
In the one photo that you provide your chin looks symmetrical. I can see some strain of the mentalis muscle that can be treated with some Botox. Good luck