Currently I have issues with hiss and a deep background hum, indicating bad shielding of the tonearm wires and some grounding issues.
@reimarc You need to figure this out before worrying about rewiring the arm or dealing with connectors! Otherwise you may find you have the same problem after all that work! Based on other comments in your opening post, I suspect this will be the case.
FWIW, the ground wire does not have to be carried within the shield. It might help to explain how tonearm wiring actually works:
The cartridge is a balanced source. Like any balanced source, the signal is carried in a twisted pair of wires to the input of the preamp. It does not reference ground, hence the ground wire is merely to connect the shield (the arm tube) to the preamp chassis ground. On this account, the actual shield can be around the signal wires only.
However, you have a foil shield which is great at RF frequencies but bad bad not good really terrible for audio frequencies. Its no surprise you're having hum problems. The 'shield' is picking up noise and injecting it into your audio wiring within. So you'll want to rethink your tonearm cable, not your tonearm wiring!
Since you are likely running the system single-ended as most people do, its acceptable to shield the '+' (plus) output of the cartridge with the '-' (minus) output once the arm wires have exited the tonearm, but this should not be the same as the ground wire connection. That's about the best way to connect a balanced source to a single-ended input and is why tonearms are the only 'single-ended' source that needs the ground wire- in fact its not a single-ended source.