It's been a while since I have had a 424 MkIII, but if I remember correctly, you should set the recording for each track to "direct".
I don't have and I never have had a Studio 24, so I'm not sure it would work, but because it has both "group" outs and "l/r" outs that are selectable by track, you might be able to employ the two to act as a virtual 4-buss mixer. That is, run all of your mics, lines, etc into the mixer as normal. For stuff you want on track 1, pan left and select "GRP" near the fader. Deselect "L/R". For track 2, select the same things but pan right. For track 3, deselect "GRP" and select "L/R", then pan left. For track 4, pan right. Then run the GRP 1 output to input 1 on the 424. GRP 2 output goes to input 2 on the 424. L and R main outs go to 424 ins 3 and 4, respectively.
Also, make sure that both buttons near the GRP faders ("mono" and "Link to L/R") are deselected.
That's just a guess, but it might be worth a try.