Drills are important, but if the instruction is not there then maybe the school is not up to scratch. You should read some of the SBI stuff about aliveness and compare that to what happens at your school. Drills should be progressive and challenging. As you progress in a drill you should eventually have a feeling of "XXXX me if I mess this up he will take my head off".
Some boxing gyms (especially pro gyms) won't have classes and people will either train alone, pay for a coach or get picked up by a coach. Even in some amateur gyms you will have to go down for a while until your considered serious and then get some coaching. Maybe this is different in the States.
If you training with MMA be careful as some gyms, especially boxing gyms, are very set in there ways. They may not be happy with you making adjustments to how you train (i.e widening your stance, reducing you ROM in ducking and weaving etc).
Talking of kickboxing for fitness my girlfriend has just been training with me, running across the garden, throwing a combination on the pads and then running back. Afterwards she said "I really liked that, it was like It's A Knockout".