Could impedance mismatch explain my distorted signal?
Here's the schematic of your gain and scaling circuit that you hid in an external image:

You don't show any connection between the ground of the DAC circuit and the ground of the amplifier circuit. This could lead to the inputs of the amplifier being well outside the power supply rails.
When you connect both the DAC and amplifier circuits to the scope at the same time, you also connect their grounds together, and eliminate this problem.
You can make the system work without the scope present by connecting the shield of the coaxial connector on the amplifier input to the amplifier circuit's ground. Possibly you could do this with a resistive/capacitive connection rather than a direct connection, depending on how far apart these two boards will be in your system, what kind of shield currents you expect, etc.