You could do it with texture paint, you would have control on how your spots appear exactly, or, as John says, you could do it procedurally.
If you choose this second option, here is a try. Set this nodes chain: Texture Coordinate (Object output) > Mapping node > Texture node (here Musgrave) > ColorRamp > Mix Shader. Mix your 2 colors (carrot color and spot ...
I would recommend establishing a paint set up in Blender first on the model by painting details on the model in the 3d view using Texture Paint mode, and then take the saved image textures into Photoshop to further build on the initial block out. Here in the picture I have used a cloud texture on a texture paint brush set to multiply and Rake and Random for ...
Without seing your shader setup, it is difficult to provide a definitive answer, but I will try based on what you have provided so far.
In the shading tab, add a new node for texture coordinates and wire it into the
normal of your image texture.
Blender needs to know how to apply the image, you have specified UV unwrwap, so you need to tell the shader ...
Most definitely you can mix your vertex paint and your texture paint together in your material node tree and preview the result - here is Suzanne with some default Dirty Vertex Paint and a flat brown image texture slot mixed with a color mix node set to 'Overlay' - left window is the vertex dirt, right window is the combined shading result.
Craig D Jones is almost right, the answer is Rake.
Both sculpting and UV painting both have the same option that will do what I'm looking for.
The Rake option when checked makes the pattern follow the brush direction.