Search TB for HaVIC5's youtube videos on melodic phrasing. Or he will more than likely repost them in this thread
Right on cue...
This video actually might cover things you already know, but still don't know how to go about internalizing it. The main thing I'd do is what you suggested....transcribe! Miles Davis solos off of "Kind of Blue" are considered a gold standard for the beginning transcriber, but you can also transcribe Paul Desmond (off of Dave Brubeck's album "Time Out") Lester Young and even Horace Silver for other very "accessible" solos that still swing like mad. Even on the so-called "simpler" solos, it will take a LONG time and a lot of patience to get all the notes and rhythms down, never mind all the nuance. I recommend never writing a transcription down until you've memorized how to play it on your instrument. That way, you're really connecting what's going on with how you play it on the bass.
Once you've gotten down the solo, however, write it out, and then proceed to analyze the crap out of it. This might be hard if you don't have that much of a theoretical background, but descriptive analysis is just as potent. Once you analyze every note for its function on the chord (passing tone? tension? chord tone? something else perhaps?), look at the phrase structure. Ask yourself, how long do the phrases last? What is the primary rhythmic pulse of the phrase (quarter, eighth, sixteenth?) What are the antecedent/consequent relationships between two phrases? What is the melodic contour of this phrase (rising/falling?) and what might that contour do in the antecedent/consequent relationship? How long is there rest between phrases? In a stream of constant eighth notes (sixteenth notes) can you break that into smaller phrase units? What is the motivic content of the phrases, and how, if at all, is it developed? Does it reappear later in the solo? As the solo progresses, how do the relationships between phrases change?
You can spend a lot of time with questions like these analyzing a solo, and the more questions you think of asking yourself, the deeper in it you will get. Once you've done this, start taking phrases of the solo and playing them verbatim (or not) in your solos. You'll quickly find what works for you (what you like) and what doesn't (what you don't like), and as soon as you figure out what you like, you'll have just added to your vocabulary. Even if its just a few notes out of the solo that you hold on to, that's plenty, because the ear training and analytical process will have contributed significantly to your musical awareness. Its a long, slow process, but its worth it.