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Redesign website using new template

I created a website based on a template. We are now one year further and we need a new website based on a new template. Now I’m a bit confused about what the best way forward is?

Should I build it on the new site? Can a move overall purchases (hosting CMS etc) to the other site?
Should I build it on the current site? Won’t that destroy all the nice class names etc?
Also, I have a CMS in my current one,… can I copy that over?

Any tips experiences on this would be much appreciated.