You are picking up noise on the long input wires. Look into what Skeptic2 suggested.
As for resistance, don't be afraid, it's just a resistor. I gave you the hint that you need long body resistor. When I did the transimpedance amp with high value resistor, I used DIP package and I bent the -ve input up and solder the resistor on the pin up in the air. You need very good cleaning spray, compressed air to blow away the solvent right after cleaning. 1G is not that high, you do this, you'll likely be fine.
If you measure down to pA, watch out for vibration. I worked on Mass Spectrometers and the vibration from the turbo pumps introduce vibration noise. We end up using rigid coax for input and mount the amp on the frame to lower the vibration to the amp.