I had already pointed you to section 7.13 of the data sheet, but your design is missing the series resistors:
Proper stability and frequency response are achieved by connecting 10 μF load capacitors at the regulated output pins. The optimal configuration is achieved by connecting a small resistor in series with the capacitor in order to guarantee the controlled ESR required to ensure the proper phase margin, together with the best performance in terms of noise and PSRR.
Like most LDOs, the load capacitors in this device require a certain minimum ESR value. Ceramic capacitors (MLCC) usually do not have this, so you have to insert the series resistors seen in the EVB.
In the circuit of the EVB, however, you can also see some 0-ohm resistors, which of course you don't necessarily have to insert, they rather serve the purpose of being able to easily insert them into the EVB later on, if necessary.
In conclusion, no one can say that the STuW81300 will work correctly if these instructions are followed. There are many more important things to consider: RF design and layout, board material if necessary, crystal parameters, type of passive devices etc etc. For this reason, an STuW81300-EVB is provided with which to make initial investigations and comparisons.
But at least you would have laid the foundation for hopefully low-noise and stable LDOs.
Good luck!
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/Peter