Google doesn't care much about what tags you use these days. Historically Google put extra weight on words in header tags because Googlebot only looked at the source code of the page and could pick those tags out. Today Googlebot renders pages to see what words are prominent on the page. I wouldn't try to include
h3 tags just for SEO. I'd use those tags because they structure the data on the page and make it easy to style your headings in ways that are appropriate for users. Google will give extra weight to the text in the headings when they see they are rendered prominently for users.
Using header tags inside paragraphs isn't going to work well. Those tags are block level elements and will cause different text sizes, line breaks and extra white space within your paragraphs. If you use CSS to style the text in the headers so that it looks just like other paragraph text, you lose all the possible SEO benefits of using those tags. In addition, it might actually look spammy to search engines and open your site up to an over-optimization penalty for stuffing keywords.
In all likelyhood you can just remove headings from your site without hurting your SEO. I wouldn't try to stuff headers inside paragraphs in an attempt to preserve rankings when doing so.