Paid full domains is better than a free sub-domain name.
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As stated above paid domain are much more far better than blogspot. Do buy a domain and start working on it with fresh content.
Rest as if your prevous content is useful and necessary then redirect the particular url to the new one as this redirection doesnt consider as duplicate issue of content and you will be able to work freely on new domain.
I've heard that Google has slight bias in ranking blogspot over other domains (due to google service providing to another google service, it even makes sense) so it potentially COULD harm your traffic.
You can buy the domain name if you want but if you want that search engine will find you then you need to done some link building and seo for your website choose the best keywords which you want use and then after do some work you will be index on search engines.