Reasons why DNA doesn't determine traits

Multiple different information layers in the cell respond to environmental signals

1.  DNA sections (sequences) are never used directly by cellular mechanisms. Parts of the DNA are always read into a template RNA strand (transcription) before further processing. The template RNA is then modified by several complex mechanisms, such as alternative splicing machinery. The mature RNA-product is regulated by several epigenetic factors and mechanisms before final protein production. 

2. DNA genes need to be told at what level they are activated or silenced. This is regulated by several mechanisms and factors, such as chromatin structure and folding (3D genome). Histone epigenetic markers affect strongly the folding and shape of chromatin. Chromatin structure is organized by complex motor protein called condensin. It is able to form loops so that certain parts of the DNA can be in flanking state regulating the activity of transcription

3. Histone code is a biological database. Histones are structural proteins of chromosomes. Histones pack and order DNA into structures known as nucleosomes so that it fits within a cell’s nucleus. Each nucleosome contains two subunits, both made of histones H2A, H2B, H3 and H4 – known as core histones – with the linker histone H1 acting as a stabilizer. Many chemical modifications can be found in the tails of the histones. These include, but are not limited to acetylation, mono-, di-, or tri-methylation and ubiquitination, and can occur in different amino acids in the tails of the different histones.  
Any real histone code has the potential to be massively complex; each of the four standard histones can be simultaneously modified at multiple different sites with multiple different modifications. To give an idea of this complexity, histone H3 contains nineteen lysines known to be methylated—each can be un-, mono-, di- or tri-methylated. If modifications are independent, this allows a potential 419 or 280 billion different lysine methylation patterns, far more than the maximum number of histones in a human genome (6.4 Gb / ~150 bp = ~44 million histones if they are very tightly packed). And this does not include lysine acetylation (known for H3 at nine residues), arginine methylation (known for H3 at three residues) or threonine/serine/tyrosine phosphorylation (known for H3 at eight residues), not to mention modifications of other histones. (https://en.wikipedia.org/wiki/Histone_code)

Histone markers are regulated and maintained by long non coding RNA-molecules that have a crucial role in development of embryonic cells. Histone markers affect strongly traits of organisms. For example skull morphogenesis is regulated by histone deacetylases. Fur, coat and pigment colors are also regulated by histone markers. And thousands of other traits...

4. DNA methylation is involved in the regulation of several cellular processes such as chromatin stability, imprinting, X chromosome inactivation and carcinogenesis. It strongly affects alternative splicing and RNA polymerases. Methylation stabilizes the genome and disrupted or aberrant methylation patterns are often the reason for genetic errors.

5. Non coding RNA molecules are transcribed from the DNA by very complex mechanisms called RNA polymerases I and II. They are not direct copies of the DNA because of complex modifications. Their job is to transmit epigenetic markers (over 140 different types) for the DNA, histones and other RNA:s. In this way cells are able to get information from surrounding environment. RNA-molecules are often packed into transporters called extracellular vesicles.

6. Transcription factors are proteins produced by previous mechanisms 1.-5.

The cellular information can be categorized as follows:

A. DNA - 
a passive digital information layer, a library for production of RNAs.
B. DNA methylation - analog information layer, meta-data
C. Chromatin structure and folding - analog information

D. Histone markers - a digital database  CONVERTED to chromatin folding (A/D converter)
E. Epigenetic markers of RNA - both digital, analog and meta-data
F. Transcription factors - meta-data

All living organisms receive signals from surrounding environment. This means diet types, climate, stress factors, sensory stimuli, toxicants etc. Any change in organisms is due to epigenetic regulation of existing biological information or gradual but inevitable disappearance of information. That's why there's no mechanism for evolution. Don't get lost.