I'll give you an example why it isn't as simple as sex and DNA.
There are examples of organisms that acquire DNA from food. Or even from parasites. And vice versa.
There's a sea slug (Elysia viridis) in the Mediterranean. It's a slug, so sexual reproduction doesn't come into the picture. It feeds on a plant, digests it, obviously, but also acquires the DNA from the plant that allows it to produce it's own food by photosynthesis. It was the only example I knew for years, but several others have been observed in the last decade. It's not a simple transfer of DNA, but the animal has the necessary supportive systems for photosynthesis in it's own DNA, which allows it to use the plant's DNA to complete the photosynthesis process.
There are numerous examples of parasites using the hosts's DNA in all sorts of ways. To avoid detection by the hosts's immune system. Or to take control of the host's behaviour. And sometimes, the host's DNA leads to a new species or sub-species of parasite.
And since there are no known species without parasites, these DNA transfers play a significant role in evolution. In fact, most parasites have several DNA transfer systems aboard.
The last few years a number of other pathways for DNA "acquisition" have been found. None of them has got anything to do with sexual reproduction.
I won't even start about fungi, as that would sound far too "Avatar" for closed minds. We already know mycelia transport signals and even food or medicine between other species. There's ongoing research to see if it also transports DNA, or even basic amino acids that get somehow reassembled into new DNA. Research like this shows that evolution has far more aspects than simply "the survival of the fittest".