I was able to find the total current in the following circuit (.145 mA). I also was able to find the voltage across the first capacitor (7.71 V) and also across the second capacitor (2.29 V). But I can't figure out the voltage across the rest of the capacitors.
I know that conservation of current does not apply to circuits like these.
I'm not entirely sure what you mean by conservation of current, but Kirchoff's current law certain applies here, as does Kirchoff's voltage law.
You're doing "steady-state" AC analysis, so start by computing the complex impedance of each capacitor at 300Hz. Then solve the circuit using your favorite circuit network methods with these constant impedances (nodal analysis, mesh analysis, etc.).
Since the circuit is composed entirely of capacitors you can use the same techniques as with a resistor network, except use \$\frac {1}{2\pi f C_X}\$ for each of the resistances (and each will be in ohms).
The actual impedances are all purely imaginary (and negative), but that cancels when you calculated the divided steady-state voltages. The input current will also be correctly calculated, but keep in mind the phase will be 90° leading wrt the source.
To do it systematically for steady state, you can apply KCL and KVL and solve with each of the complex impedances, and that will work in all cases, not just this special case.
