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I am a bit confused with your post, especially this part:
I've been blogging twice a day for three to four years now and suddenly about a month ago, every blog post URL was being seen by Google as a 404 error. I started off by deleting each post, then having to delete the page entirely. Nothing. Google could still see it.
These two bolded parts written as they are seem to be contradictory. Reading your post again, this is what I am getting from it:
It seems that the blog pages started to return 404 (not sure if they suddenly started to return 404 or did you do something on purpose to return 404).
Then you decided to physically remove the pages, but I am confused what do you mean by "Google can still see it" since you previously say Google is geting 404 response. Do you mean the content is shown, but the response is 404? Or do you mean the URLs still appear in Google search results? Reading on your post and assuming you want to remove these blog pages from your site, then:
You should ensure that pages return 404 (which they do, it seems). You should also ensure you are not internally linking to these pages (which you did, using the URL link tool). And you should remove these pages from the sitemap.xml (which you did). So far, fine.
Once pages were removed, Google will report them in WMT as "404 errors" - this is normal
. You have checked 404 errors "linked from" information and you saw that the referring page is your sitemap.xml, but the sitemap was updated and resubmitted. Note that it may take some time for Google to process the sitemap AND to re-crawl the page after processing the sitemap to remove the reference from the sitemap to your page. So you need to give it a bit of time.
To me everything seems fine so far. Here is some additional info on 404:
When the page starts to return 404, it will still be shown in Google search results for a while, and then it will eventually drop from Google search results. It may take some time for Google to drop these pages from SERPs. To see if a page is still in Google SERPs, you can use the following search command:
site:example.com inurl:blog.php (replace example.com with your domain name)
If pages keep returning 404 then the above command will return less and less results as the time passes.
Removed pages will show in WMT as 404 errors, at least for some time. If the page has external links, it may keep coming back in 404 errors report of WMT even though you acknowledge errors in WMT. If there are no external nor internal links, page may still come back in 404 report, but this will come less and less often since if the URL keeps returning 404 for extended time, Google will crawl these URLs less and less frequently. Just keep acknowledging these "errors" in WMT.
You need to give it some time for 404 pages to disappear from Google SERPs as Google will need to re-crawl each page to see 404 response, and it may expect 404 a few times from the same page before it drops this page from its index (and SERPs).
If I haven't got right what you are trying to do, can you please try to explain better what your problem and question is.