Could this power supply be causing problems?
Problem: no decoupling on power supply.
Solution: solder a capacitor on the ADC SMD adapter, preferably &gt;100µF with ESR below 1 ohm. Clip the leads as short as you can. LM358 would also need one, any electrolytic will do. You could use ceramics, but since the power supply is quite far away, you'd need a large value to compensate for the wiring inductance, and you already have electrolytic caps in stock.
Problem: SPI in flying wires.
I see 3 SPI wires, with no ground return, so the high frequency digital currents return through the wires to the power supply, which means the bits will fall off and you will get wrong readings. So you should add a ground wire between the ADC and the FPGA, twist everything together, and slow down the SPI clock a lot, and add source resistors on all outputs to slow down the edges, something ridiculous like 330R-1 kOhm, until you get clean edges on all signals on the scope.
Personally I wouldn't use a solderless breadboard to experiment with your ADC. SPI in flying wires is always going to be a problem. However, it is absolutely possible to send 5ns rise time signals in handcrafted wires... if they are taped on a ground plane and source terminated. Note 50MHz SMD oscillator dumped in the middle of the board, no problem.

Problem: power supply noise
I's difficult to know what causes this without further tests, but you could remove the ADC from the breadboard and measure the power supply noise and ripple with the scope. You can also add a dummy load resistor to draw say 10mA on 3V3, see if that improves it. Some power supplies kinda go to sleep (ie, loss of transconductance in the pass device decreases open loop gain, which worsens regulation) when the load is too low and thus voltage ripple increases. But it could just as well be noise from the FPGA coupling through the flying wires. So you should experiment, also disconnect the FPGA, start from blank slate and add each part back, see when the power supply misbehaves.