I got deeply involved in web design in the mid-'90s and spent most of my time over the next 25 years involved in the very things you’re asking about. About five years ago, I decided that I was done with it and decided to focus full-time on what I always liked better, which is print.
So with that perspective in mind, I’ll respond to your question.
A Bachelor’s Degree in Graphic Design is a huge advantage, but you already know that. The subsequent experience you mentioned is also important. Your portfolio of design work, which I haven’t seen, is even more important.
You expressed an interest in the types of sites found on awwwards.com. However, your direct experience is limited to drag-and-drop online web-building tools. Unfortunately, the sites you’re intrigued with were designed and coded from scratch, so your experience with Wix and Squarespace doesn’t get you very far.
UX/UI design has gotten too specialized and team-oriented for me. I always preferred designing and coding my own sites and apps. Even when using a CMS, I’ve always preferred designing and coding the templates or themes myself. I have a very hard time separating HTML/CSS/JavaScript/PHP from the design and regarding those languages as separate from the UX/UI. However, knowing how to design and build every aspect of a website or an app has gotten way too complicated and specialized into dozens of little niches. It’s too much for anyone to keep up with.
Setting aside the drag-and-drop website builders, most websites today are built in various CMS’s using various frameworks that enable a designer to pick and choose various options with a minimum of coding experience. It’s largely boilerplate type work where one’s creativity is limited by the CMS and the framework. This is why so many websites have roughly the same UX/UI as all the others.
The sites you’re interested in weren’t built that way. Most have all the looks of a CMS. However, I’m reasonably certain that the themes and templates of the most interesting sites were designed and coded by teams of people who built the sites to be what they wanted them to be instead of working within the parameters of commercial templates/themes.
My firm opinion is that in these kinds of teams, each specialist needs to have a good grasp of all aspects of the project. For example, a coder needs to understand and have a reasonably good feel for design. Likewise, a designer needs to understand what can be done or not (easily) done within the restrictions and possibilities of the development space in which the project exists.
As an analogy, let’s take architecture. An architect might be primarily concerned with building a functional and beautiful building. However, that architect also needs to know enough about engineering, building codes, materials, carpentry, and other things to design a building that’s practical and cost-efficient. Much the same is true of UI/UX design; the designer needs to know enough about the technical aspects of the job and the coding to design a site that’s practical for the coders and programmers to build.
Software applications, like Figma, Adobe XD, and others, smooth the awkward barriers between UI/UX and coding, but, in my opinion, everyone involved in building a website or an app still needs to know the basics (and more) about the other aspects of the job, in much the same way as a specialist physician needs to be well-grounded in basic medicine before specializing.
You have the design degree covered and some design experience under your belt, so if you’re portfolio shows promise, I see no reason why you can’t pursue UI/UX design, but you’ll be starting at the bottom.
To maximize your chances of getting your foot in the door, you could study all the basics of how websites and apps are built. You might even take it upon yourself to design and build a basic website in WordPress and learn some HTML and CSS. The more you know, the more likely it will be that an employer will be interested. Most employers want to hire people who can be useful right from the beginning without the need to train them from the bottom up.