I think @Aaron is on the right track in pointing you to the Wikipedia article on yod dropping and yod coalescence. However, some clarifications are in order.
There is actually a family of variants for what you are calling the "softer d"--the variants can respectively be transcribed in IPA as [dj], [dʲ], [d͡ʑ], and [d͡ʒ]. Remember that in IPA [j] (called yod) stands for the sound at the beginning of the word yell, and the symbol [d͡ʒ] stands for the sound at the beginning of the word gel. When you used the letter j in your question, I'm guessing the sound you had in mind was more like the latter, although by placing the letter d before it you were perhaps approximating one of the "intermediate" sounds ([dʲ] or [d͡ʑ]). I refer you to the Wikipedia article on IPA and the relevant linked articles (you can click on individual symbols in the IPA chart for in-depth explanations of them) for a review of the sounds represented by these symbols. There you can also read about the differences in articulation among these different sounds, but generally speaking the articulatory differences between [d] and the variants mentioned here have to do with the placement of the tongue immediately after the closure (farther back for the variants than for the [d]) and the speed at which the tongue blade moves away from the roof of the mouth (more sluggish for the variants than for the [d]).
To discuss the transcriptions in terms of a concrete example, some speakers pronounce dew as if they are pronouncing the word yew with a [d] sound before it. That pronunciation would be transcribed as [djuː ]. Some speakers of the dialect (or dialect group) to which you are referring pronounce it like the word juice without the [s] sound at the end of it, i.e. it is indistinguishable from the word Jew. We can consider these pronunciations to be at opposite ends of a sort of spectrum, where the first pronunciation comprises a [d]-[j] sequence and the second one comprises a single sound (considered a coalescence of two separate phonemes /d/ and /j/). Yet other speakers might pronounce it somewhere in the middle, which would be transcribed with one of the other variants mentioned above. (I myself do none of the above; I pronounce it just like the word do, which some would analyze as yod dropping.)
Finally, it is important to point out that by using the word phoneme you are presupposing that the speech sound (or sound sequence) to which you are referring is the realization of a single underlying unit. Chances are you just meant speech sound. However, this brings up an issue that is relevant for @Aaron's comment about the pronunciation having a rule-based--as opposed to lexical--source. The existence of the minimal pair dew vs. do (as well as other relevant pairs like twos and Tuesday) in the relevant dialect(s) shows that the environment for this sound (or sequence of sounds) is not predictable and therefore the distinction must be lexical.
