The issue is that SWF itself isn't a video format, although you can embed a video into a SWF. If the SWF actually contains the video (as opposed to being the video playback UI that loads an external video, which is also common), then you'd need to extract the video data into a separate file and then do a video conversion on it. There are third-party tools that exist for stripping video data out of a SWF, but I can't vouch for them, and it sounds like you haven't had a lot of success.
If you're lucky and have external video files in an older codec (FLV, etc), Adobe Media Encoder should be able to convert them.
One good way to figure this out would be to load the page while looking at the network tab of the browser's developer tools to see what's getting loaded as the page runs. If you see separate video files loading when the video is playing, that makes things much easier.
If you only have SWFs with embedded video, do you have any of the source files (FLAs, etc)? Did you create the SWFs from another tool like Adobe Captivate or something? If you have your old Captivate projects, current versions of Captivate will let you output directly to HTML5 and JavaScript.