I took the old SAT with essay and scored a 10 both times. I'm happy with these 10's and don't want to relearn the new SAT essay style. Can I take the new SAT without the essay and report my old SAT essay scores with it?
You won't be able show colleges just your old SAT essay scores and new SAT multiple-choice scores. I assume if you plan on taking the new SAT that you weren't satisfied with your multiple-choice scores on the old test. If you want colleges to see your old SAT essay scores, you will need to report your multiple-choice scores as well. You can't break up the sections.
You can still take the new SAT without the essay if you're ok with sending your composite scores from the old SAT. However, there's no guarantee that colleges will consider multiple-choice section scores on the new SAT more strongly than multiple-choice section scores on the old test (or even at all, since it's not technically a complete test without the essay). You should check if the colleges where you're applying superscore across the two versions of the exam. If they do, it's more likely that they'll consider your section scores on the new SAT AND your old essay score. Read this article for more information.
Overall, I'd recommend calling the admissions offices at the schools where you plan on applying and asking them about their policies. The transition to the new SAT has created somewhat of a divide amongst colleges as to what they will accept. You should get your information directly from the source so that you can give yourself the best chances of admission.
Disclaimer: This answer assumes you're applying to schools that expressly require the SAT essay. Make sure you check specific admissions policies! If the new SAT essay isn't an application requirement for your chosen colleges, then you're totally in the clear.