Not sure if I understand your question correctly.
To search for a string wrap it in quote signs, as @JMichaelTX has shown. This should work that way among all site searches and among all major search engines (well, I noticed that certain engines have started to ignore the quote marks, above all Google, but also the better ones, like DDG).
To do a search on the page I usually just invoke the browser’s page search with ⌘F.
But, somehow your post #2 tells me that the problem is a different one. Unfortunately I don’t understand it.