If I read you correctly you want to solidly mount your back and side panels of plywood to the legs which are solid wood.
If that's what you're doing, it's sort of a no-no, but people do it. Biscuits or pocket screws would be easiest. You might find that it's easier to make a plywood box and then overlay the corners with the oak wood. This takes all the headaches out of the equation but still looks good.
You can certainly dowel the stretchers for your drawer instead of m&t. Use a dowel of the same sort of wood. I do it all the time with a 1" stretcher on a 2" leg. For a 3/4" stretcher, a 3/8" dowel will work and is readily available.
I'm not entirely certain as to how you're making your panels. If you could describe them in detail I expect you'd get several ideas in response that might save you some frustration.