core integration

Not sure exactly what you mean and I'm mostly writing because I don't think I really engage my core when I swim. Maybe that explains my swim times...

The key for me for my swim stroke and timing is all from the hips. My hip rotation is where the power comes from and in many ways is similar to the concept of a golf swing. I think having too much focus on the core for powering the swim could take away from the kick and hip interplay.

In the sense that I'm always keeping the body aligned whether rotating to the other side or staying whre I am for the moment the core is engaged to keep a neutral pelvis and a firm posture. If you let it all go between rotations there is a tendency (for me) for the hips to float and the belly to sag causing a swayback that's not good for streamlining. It's similar to standing straight up with good posture. You're not 100% relaxed, but youre not holding tension either.

You shouldn't think about engaging your core. If you have problems with fish tailing (where your legs swing side to side for breathing) there are two things that come to mind: cross over; you can use an ankle lock to swim with a buoy and it does wonders for your form.

If your hips turn at different times than your shoulders, same thing

Just re read your question: if you're having a hard time turning you just need to practice your body roll. You turn as one unit, but your body doesn't completely turn-just a little rotation enough for you to turn your neck to take a quick bite of air.

But you shouldn't feel your core tighten up, swimming is a very "relaxed"sport

Another poor result of letting core relax can cause people to disconnect upper body with lower body. That makes the swimmer almost snake or zig-zag through the water, which is undesirable. I think the analogy to standing up straight is very good. A kind of stiffness without being contracted/flexed