They both seem to be widely used. There is the variant "at a time like this", which is clearly correct, but I'm curious about the mismatched "times like this".
Your titular question is "Are [they] ... both correct." Your later question narrows it down to "times like this". I'll focus on the latter.
As usual, it depends what you mean by correct. If you mean grammar, then clearly there is nothing wrong with "times like this". There could be many times that were or will be like 'this time.'
"I have had many Mondays like this."
If you are asking about frequency then Google ngram is your friend.
You can see from the graph that "times like this" occurs marginally more frequently than "times like these".
Not only does ngram give numerical statistics but you can also click on the links at the bottom of the page to find examples.
I agree with Chasly. A native speaker might switch between these options depending on the emphasis desired. For example, if the house was burning down and my mother asked me if I liked her new dress, I might say "How can you think about a dress at at a time like this!" "A time" and "this" emphasizes the specific moment. In contrast, a politician giving a speech about general social problems would probably chose "In times like these, it's important to pull together as a society." Personally, "times like this" sounds awkward to me and I wouldn't use it. But that doesn't mean it isn't commonly used.