Generally, the time requirements are upon enlistment, not completion of school, but I was not Navy so they may be a little different. With my career field, though, all the Navy people I trained with (joint school) had the same deal... 6 years from time of starting basic training, not 6 years from completion of school. Verify that though. I also found that while 6 years was "encouraged" none of the jobs I looked at required it. They give you extra stripes when you finish basic training (or while in basic training depending upon what service... not sure of Navy) so you get higher pay too, but nothing I looked at required 6 years. Again, just verify that with the recruiter (and the Navy diver you talk to).
As for going to college full time, I highly doubt that unless you're very motivated. Generally speaking, taking 2 or 3 classes at a time is not too difficult and taking classes is generally encouraged these days, especially if it's related to your military career. Taking a full course load is often very difficult to schedule around your military workload though. Not impossible, but very difficult unless you're doing self-paced or online classes. On float there is a higher possibility of doing those types of courses though.
As for going to college full time, I highly doubt that unless you're very motivated. Generally speaking, taking 2 or 3 classes at a time is not too difficult and taking classes is generally encouraged these days, especially if it's related to your military career. Taking a full course load is often very difficult to schedule around your military workload though. Not impossible, but very difficult unless you're doing self-paced or online classes. On float there is a higher possibility of doing those types of courses though.