I think that's plenty of notice. Companies don't give you any notice when they fire you or lay you off, right? Two weeks' notice is standard professional courtesy. You've given a little less than that, but I think that's fine.
Taking your sick time, I feel, would be unethical (unless you are actually sick).
Remember that you may need your current employer for a reference someday, and even if not, you never know who's in your managers' network.
BTW, I suggest you use any benefits you are still entitled to from your old job. If you have health insurance, now's a good time to go to the doctor, dentist, eye doctor, etc. - especially if you won't have insurance at your new job right away. If you have tuition reimbursement, flexible spending accounts, wellness benefits, etc. - use them.
Good luck with the new job!
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