Does Ohm's law apply in an RLC circuit?

The place where beginners usually screw up with Ohm's law is by not being clear about which voltage and/or which current they are referring to.

Ohm's law relates the voltage across the resistor to the current through the resistor.

"V" in your diagram is not the voltage across the resistor; it is the voltage across the capacitor.

The voltage in your figure is not the one across the resistor, so it'll not work as intended and that yields incorrect results.

Remember as well that for capacitors and inductors, the use of phasors or the Laplace Transform may ease all your time domain calculations, a quick grasp of MATLAB functions can solve you lots of equations for bigger circuits, which would take lots of sines and cosines out, effort cutback and will leave only what is actually important in the end.