Depends on how embarrassing something is. I've gotten much better at expressing myself over time, so I clarify stuff much more often now when something does feel embarrassing to me, which isn't often. Sometimes I'll just hold in the feelings or put them behind bigger feelings if I really want to do something and need to push past the embarrassment.
Most people get embarrassed, so you shouldn't feel bad about it in the slightest. Everyone makes mistakes, so when you make yours, just accept them. If people give you shit about embarrassing stuff in the past in a malicious way, then they're bullies, and you should either leave them in silence or just kick their ass(es).
Embarrassment and shame have plagued me all my life, and I'm 31 now and still have trouble dealing with those feelings. I'm currently working on it with a psychotherapist. In my case, the embarrassment itself is relatively benign, the issue is just that it tends to lead me to negative thoughts about myself, such as "I'm stupid and fundamentally defective", which cause me to engage in harmful activities. Those thoughts are often irrational. The therapist has presented me with a sort of mental framework for analysing these thoughts and accompanying feelings, as well as the actions that trigger them. It's very useful to me. I'm still working on getting into a habit of using it. It has me specify the action that happened, the beliefs that arose in my head as a result, the changes in my physical condition (bodily responses to emotions), and the consequences of all the beforementioned, i.e. how I reacted. Then, I can confront my beliefs if they are irrational, and examine my reaction. It's a very versatile tool for introspection.
Been embarrassed so many times that I don't think it would affect me anymore
If it's small I just kind of make it a joke and that usually makes it bearable
The ones I can't do that on are usually so bad that I just kind of wait for it to pass and just accept the outcomes
Some years ago, I bought my first ever car. I had only recently learned to drive, and I wasn't good at it - I'm always slow to learn new things. The truth was that I was really really insecure about my driving - I was having nightmares frequently about being forced to drive on dangerous roads because of emergency situations.
On the test drive, with the seller in the passenger seat, I almost got into an accident when I tried to forcefully merge into a busy lane. I swerved out of the way of a car at the last split-second and ended up being forced to take the highway out of town. We sat there in silence. Eventually we turned around and headed back, and he tried to play it off like nothing had happened, so I followed suit. I told him the car seemed good, and we scheduled to meet later for the actual purchase at a garage.
I went home and locked my door behind me, went into my bedroom, locked my bedroom door behind me, and had the most intense panic attack of my life where the moment of the car I was merging into looming closer played on repeat for about 3 or 4 hours until I passed out. I woke up later and continued to freak out some more, and then I was berating myself about how I can't do any normal thing correctly. During that moment I was like "wait - I had a close call, not even a real accident, and it's completely debilitating me. I can't live like that. I really can't. I need to get over this immediately." Somehow, that worked.
For me, it's somewhat similar. I think about everything negatively, and it's hard for me to imagine how I want to be or want to feel, but easy for me to imagine what I would consider awful or disapprove of. I focus mostly on the consequences of those feelings.
"Is this useful to me? What is this feeling actually doing to me, for me? What is my plan? If I intend to keep trying, to live and grow, then I should hold myself back only the minimum necessary amount."
That helps me see it as irrational, as not 'right', and that gives me a huge boost in my ability to move on from it. It will still haunt me somewhat, but somehow I'll know "I can get over this - eventually." That's what was sort of forced into place by that driving incident - my embarrassment and the subsequent response to it was so severe that even in the middle of the panic attack I realized how absurd it was.
What you did after you came home resonate a lot with what I would like to do once I remember some cringe moments in my past.