DW4 created sites

Can you share a link to the site to see what we are talking about?  As other's have mentioned, HTML is HTML and it would be editable in a later version for that reason (although some UI features might no longer exist).  I don't think there is any time to buy here and that your best plan is to figure out a transition forward whether that means coding a new site, transitioning to a CMS, or going to a website builder depending on the complexity and functionality your site requires.

A website created in DW CS4 is just html, css and maybe a little javascript.

There's no "DWCS4 HTML". What I mean is, there's nothing proprietary about the code itself. Any code editor can make changes to a site created in CS4.

Obviously certain things, like DW's Behaviors, won't be editable from the dialogue windows that DW uses, while inside another application. However, their code is just javascript. If you understand the methods used by Macromedia's development team to write Behaviors (even in the latest version of DW, Behaviors are still made from the ancient MM code), you can modify them in any code editing program.

DW's .DWT Templates and Library features won't function outside of DW because other applications don't understand what the Templates html comments mean, so you would need to make changes to each child page instead of to the main Template or Library file, if you used them in the past. They still function in the latest version of DW.

Any newer version of DW will be able to edit CS4 files without issue, as long as you work from within a Defined Site in the newer version. Even ancient APDivs and APElements will still scale and drag in Design View (though NOT within the newish Live View Editor) just as they did back in CS4. The buttons themselves for APDiv and APElement have been removed to dissuade developers from using the outdated layout method though.

About the only thing CS4 had, that the "latest and greatest" doesn't (without buying a third party extension), is the database functionality that made DW popular back in the day. That has been gone for some time. Adobe had promised to bring it back, but it has been put on hold until they fix some other issues first. Any database functionality would need to be written by hand.

As Nancy mentioned though, a site made in CS4 is likely to be very outdated and extremely unlikely to work in mobile devices. It would probably be best to start over with newer programs using more modern layout methods.

Don't delay rebuilding the site.  Dreamweaver 4/Ultra Dev came out 20 years ago.  Anything that old is woefully outdated and not suitable for use on modern web devices/browsers.