Many people are shy. Shy folks are not only easily embarrassed, they are also embarrassed by the fact that they are easily embarrassed. They tend to assume that shyness is unattractive, and that it mainly indicates lack of courage or strength. They wish they weren’t shy.
If you are shy and this is how you think, I have good news for you: shyness has important positive aspects. In particular: shyness promotes intimacy. Intimacy is a feeling of closeness, based in part on mutual vulnerability. A person who is comfortable around most folks, talking or doing most anything, finds it harder to create mutual vulnerability. How can you tell such a person is comfortable with you, as opposed to conformable with everything?
In contrast, a shy person who is awkward and embarrassed around most folks, or when talking about many topics, can more easily show that they treat you differently. If they are less awkward around you, and are willing to talk to you less awkwardly about usually-awkward topics, you can take that as a good sign that you have achieved some level of intimacy. They treat you with less caution, presumably because they are less afraid you will hurt them. In this way, a shy person can more easily show you that you are special to them.
To better bond, be shy. Reserve some topics as ones you will not talk with just anyone about, and some activities as ones you will not do with just anyone. Then by selectively talking and doing, you can signal your intimacy with some relative to others.