I agree with Colclough, organic modeling is much more difficult than placing a few primitives... You will usually start with a primitive shape (cylinder, sphere or cube... whatever you like...), but...
1. you have to spend some time experimenting with all the tools in point-edit mode to get familiar with it... Then you can modify your primitive...
2. at least 99.9% of good organic models are subdivision objects (as Coulclough said, use Build>Convert to subdivided command); that makes it much easier to get a smooth shape, which is easy to handle...
once you are able to use point edit mode and subdivision objects, you will want to learn the 'advanced' modeling... Then you will have to obey a few more rules to create really good organic models...
For example:
- almost all the faces should be 4-sided
- there shouldn't be more than 4 edges coming from one point; there should never be more than 5 edges per vertex!
- the mesh structure (edge loops, density of points...) should obey the basic rules of anatomy...
Hm... Maybe you should ignore all this for now... I think box modeling (as you said) is the best solution, just try to make it a bit more... uh... a bit more than just a sphere with 4 extruded faces!
Organic modeling is not a 5-minute work, it's not a shame to spend several days working on one model... The goal is not to have it done, the goal is to create a good model! Be patient!
...and good luck!