Nothing wrong with learning each part really well, but this frequently does not help you understand how the two lines work together.
My approach is similar to playing piano. I figure out how the lines interact and then do both as if bass and voice is one instrument. I read music and it can be very helpful to see both lines written out. Then you can see how the syncopations link together and work through them.
Start as slow as necessary so that you can actually comprehend the rhythms presented by all notes (both instruments) in order. If necessary, prioritize the timing over the pitches. Once you figure out how to cover the rhythms, it will be easier to nail the pitches (IMHO). As it starts to become automatic, slowly increase the tempo.
When you write both lines out you might get something that looks like this:
Image copied from:
Pitch (timothymichaelrolls.com) but the page really has little to do with this subject.
Notice you can see how the lines work together. Where both lines play at the same time, and where one plays a new note while the other is sustained.
Consider drawing in the pulses:
Now you see how each line relates to the other, and also how each line relates to the beat.
If you can, work through the rhythms in your head. No need for a bass or singing. Get used to the way the two lines interact rhythmically, without even worrying about plucking or hitting pitches.