Open loop op-amp white noise difference amplifier
Too much gain. The amp is open-loop so its gain will be about 100,000. Any difference between the source voltages, even 1mv, will cause the output to clip (settle as close as it can get to one of the supply rails).

Look for examples how to set the opamp gain to something more reasonable (say 100 or 1000) such as the "non-inverting" amplifier here. With Rf= 100k and Rg=1k, gain would be a reasonable 101. You will also only need one of your noise source inputs.

One warning about excessively high gain from an opamp : bandwidth is reduced (see "open loop bandwidth" and "gain/bandwidth product" aka "unity gain bandwidth") so noise from such an amplifier will not be "white" (spectrally flat). You need to decide the bandwidth you need and set the gain accordingly. For an ADC, bandwidth is usually less than half of the sample rate. For white noise, we probably don't need to be as strict as we would for general signals, but still, a bandwidth of 30kHz is reasonable given your 77kHz limit on sample rate.

For a TL082, unity gain bandwidth is specified as 3 MHz, so that limits our gain to 3MHz/30kHz = 100 for a single amplifier stage. You can choose a faster opamp, or add a second amplifier stage for higher gain. (However, it can be difficult to stop a high gain, high bandwidth amplifier from oscillating)

You may want to control the DC gain separately from the AC gain, and set the DC gain to a lower value such as 1. This is easily done with a capacitor (say 10 to 100 uf) in series with Rg. Then you can increase Rf to increase gain without the output clipping from DC imbalance (but at the cost of reduced bandwidth).

EDIT: Simple capacitor coupling would reduce the DC gain to 0. This approach has a DC gain of one, easily increased with a resistor across the capacitor, to some intermediate value.  Either approach works.

You can employ higher order filters to improve the response around your chosen cutoff point; at the moment I think it adds complexity. Or you could set the cutoff to 0.1 Hz, but I think you would have to wait about a minute before it settled after power up!