It's tough for us to guess at your capabilities.
While physics isn't entirely serial in it's development it is fairly hierarchical.
Quantum mechanics is very much built on classical mechanics. If you don't know what the Hamiltonian is from classical mechanics QM is going to be very mysterious even if you are able to complete the problems.
In general it's a bad idea to try and shortchange your studies and there is no reason to. With the math you will have you'll be able to understand CM without any difficulty.
Do you have an scholastic advisor of some sort? It sounds to me as if you need someone you can trust and can discuss these things with in person in an actual conversation.