My RC Time Constant calculations are wrong, circuit changes my input signal
In your calculations you are forgetting the output impedance of your signal generator. While usually it supposed to be 50 Ohms, reverse calculations suggest that it is about 150 Ohm. What do the specifications for DG1022 say about output impedance?

You need to start from measuring the ACTUAL impedance of your driver (DG1022). Start without any capacitor. Set 100 Hz square waveform, and record the amplitude of output WITHOUT ANY LOAD. Then connect any known load, 50 Ohms, and do the same. The difference will allow to calculate the actual output impedance of the generator.