First, welcome to the forum! And, what is wrong with your existing lens? The 18-105mm is a decent leave-on-your-camera lens, as it is. Be sure you actually need another lens!
My wife uses a Tamron 18-200mm Di-II on a D7000. I bought these for her, the D7000 itself being body-only, and offered to pay the higher price for the Nikkor 18-200mm, but she preferred the lighter weight of the Tamron, especially as she was recovering from an injured rotator cuff at the time. Plus, her employer's Nikon cameras were equipped with Tamron lenses, so she was already familiar with them. Gordon has reviewed some of these 18-200mm zoom lenses, in the review section of this site.
There is, of course, notable "barrel" distortion at the wide end, and notable "pincushion" distortion at the long end. All of the extreme-range zoom lenses have this distortion, not just Tamron. One accepts the distortion in exchange for the convenience of having one lens to cover the broader range of focal lengths. Indeed, most zooms have this distortion pattern, to a lesser degree than the super-zooms.
I should add that the Tamron 18-200mm Di-II, described above, is still available, but is an older model cpmpared to more recent super-zoom lenses, such as theones with even greater range, such as the 18-270mm mentioned in a previous post.
Notably, when I bought a Nikon Coolpix A, which is a fixed-focal-length compact camera, but with the same excellent 16 MP sensor as the D7000, and a menu familiar to Nikon DSLR users, my wife claimed it for herself. I am now saving for another one, for myself.