Hey, yeah I tried disabling all plugins but it made little difference. It could be worth changing the theme temporarily, though I think I'll have to wait until late at night to do that, as the site is workable at the moment and people are still visiting it.
It does seem that you were right though, and the problem is resources. I looked at Google Analytics and the traffic for the site at the moment is astronomical. However, almost all of the traffic is spam from Facebook! For example, yesterday there were close to 2,500 pages accessed on the page that look like this one. They are all directing to pages that don't exist and have never existed. It's very strange!
/886602052816?fb_action_ids=1491176771184310&fb_action_types=og.likes&fb_source=other_multiline&action_object_map=[831103497008868]&action_type_map=["og.likes"]&action_ref_map=[]
I think I'll have to get in touch with Facebook to see what is happening!