Hosting service for DW site

There's nothing for the hosting company to "support" with DW really. It's not like FrontPage, where additional items had to be installed on the server in order for whizbang features on your pages to work.

When they say they don't support DW, they're usually saying they won't be able to help customers figure out what they did wrong with a page, scripts, files, uploading, etc in DW.

Basically, they no longer spend resources keeping their tech support personnel up-to-date with what is now essentially dead software, which they cover with a simple "we don't support Dreamweaver" statement.

Web hosts don't care what kind of computer you have or which software you use.  That's not their job. Their only job is to take care of SERVERS & systems like email, Plesk or C-panel that run on their end.

Make a wishlist of what you want:

1. Decide what kind of site you need (static or dynamic). 
2. How much data storage do you require? 
3. How much web traffic do you expect?
4. How many email accounts do you need?
5. Which programming languages & databases?
6. Which server utilities do you need to perform routine server functions?  This can be very important particularly if you're not experienced with servers.
7. Ask if the host has 24/7 live tech support and a server maintenance package (highly recommended).
8. Does your hosting plan include annual renewal of your domain name (yoursite.com) and SSL/TLS certificates or do you need to buy those separately?  And will they install them for you or do you need to install them yourself?

These are just some of the questions you should be asking.  Depending on your situation, there may be others.  But which web authoring software you should use is completely irrelevant to them unless you're planning to run the software directly from their servers.

Most beginner sites can get all they need from low-cost, shared web hosting. Your site will share space with 20-100 other website domains. If your site is very large & gets lots of traffic, shared hosting might not be enough. You can always upgrade to costlier VPS, Dedicated or Expandable "Cloud Bursting" options as needed.

My experiences with In Motion, Hostgator and Dreamhost are all positive.  They have a good uptime record, 24/7 tech support & affordable starter plans.  Websites built with Dreamweaver & non-Dreamweaver web apps all perform well on their servers.

https://www.pcmag.com/picks/the-best-shared-web-hosting-services

Hope that helps.  Good luck with your project.

There is no such thing as a Dreamweaver site. Dreamweaver is a glorified text editor producing and using HTML, CSS and JavaScript files as well as serverside scripts such as PHP. All of these files are excepted by most web hosts.

Here are some hosts that you could try.