Shabbos, especially at the beginning, is hard to do alone.
I totally get that it can feel awkward to invite yourself over, because it’s like “Why am I forcing myself on other people?” but many Orthodox rabbis and even laymen really like the opportunity to have guests in general, and especially to introduce someone to experiences which they themselves really appreciate.
You can just find the number of a synagogue where you think you might like to connect, call up the office and talk to the rabbi or leave a message saying that you’re interested in learning more about Judaism but you live out of walking distance from the synagogue and you wanted to know if there was anyone who could host you for a Shabbos.
The worst that will happen is that it will be awkward and you won’t go back. But it could also be a lot of fun.