Op amp current noise reduced by putting pressure on signal cable
The weird thing is, the noise on the output of the current amp is
reduced if I press on the ribbon cable that's carrying the signals
from the DAC to the op amps. In the last plot, you can see the noise
is a lot less around the 4.5 s mark where I start pressing on the
cable.

Not surprising, you are introducing a few pFs or more of capacitance to the system when you touch the outside of the ribbon cable, this most likely creates a low pass filter in conjunction with the amplifier. But it would be hard to know what is really going on here because modeling your body's impedance and connection to the world would be difficult. The connection is capacitive however.
What you should do is use a shielded cable if possible. One such method of shielding would be to acquire some grounding braid and either construct a new cable with the ribbon cable inside. Use copper because the end can be soldered to much better than aluminum, the connection one end should suffice and it should be low impedance to the ground of the PCB.

Considering that the aluminum foil didn't help at all, you might want to consider a completely different wire harness, a coaxial connection that is shielded would ensure that RF would not likely affect the signal.
Another problem might be your grounding system, as common mode voltage noise  from poor grounding between two PCB's could be a problem.