Why asexual reproduction?
When you say, why don't we see more sexually reproducing species than asexual, I presume you are referring to bacteria, protists, archaea, some fungi, etc.

Most multicellular organisims do reproduce sexually, which makes sense since an organism would need to be multicellular to have cells specialized for gametes. As you stated, the advantage is that sexual reproduction offers an advantage by allowing more rapid generation of genetic diversity, allowing adaptation to changing environments. 

However, look at the single-celled organisms that reproduce asexually. Asexual reproduction is beneficial in many circumstances in that it can create individuals rapidly and in large quantities. For example, yeast can duplicate every hour under the proper conditions. For yeast, that means survival by sheer numbers. For us, it means we can make bread in a few hours.

If you believe asexually reproducing organisms are at a disadvantage due to lack of genetic diversity, consider bacteria and antibiotic resistance. The rapid generation times allow mutations to be selected for in particular circumstances. Penicillin was introduced in 1941. Two years after this landmark event, strains of Penicillin-resistant S. aureus had already emerged. Now, perhaps 2% of Staph. aureus (SA) is susceptible to penicillin, and some SA are "superbugs" with multidrug resistance: good for Staph, bad for us.

Also, it is possible for single-celled organisms to "share" DNA through mechanisms sometimes likened to sexual reproduction; lateral gene transfer occurs through conjugation, transformation and transduction. 

Thus, both asexually and sexually reproducing organisms are well-suited to their ecological niches.