Yes, this is called neuroticism.
Embarrassment as a personality trait can be measured using the Embarrassability Scale (ES), Susceptibility to Embarrassment Scale (SES), Acceptance of Shame and Embarrassment Scale (ASES), among others. A comprehensive list of scales used to measure embarrassment, guilt, and shame as traits can be found in Robins, Noftle, & Tracy (2007).
However, it turns out that many emotion traits, including embarrassment, guilt, and shame, correlate with each other in a manner that shares an underlying factor. Thus, personality inventories typically do not need to measure embarrassment specifically, as individuals high in embarrassability tend to score high in other personality sub-facets that share this factor.
In personality psychology, anxiety, embarrassment, guilt, and shame, fall under the trait of neuroticism - one of the Big 5 traits:
For example, in the Big Five approach to personality trait theory,
individuals with high scores for neuroticism are more likely than
average to be moody and to experience such feelings as anxiety, worry,
fear, anger, frustration, envy, jealousy, guilt, depressed mood, and
loneliness.
Embarrassment belongs under the neuroticism sub-facet of self-consciousness (as mentioned in another answer), and in fact, embarrassment scales are often validated against self-consciousness.
These emotions are also more broadly included in the trait of negative affectivity:
Negative affectivity subsumes a variety of negative emotions,
including anger, contempt, disgust, guilt, fear, and nervousness.
... Trait negative affectivity roughly corresponds to the dominant
personality factor of anxiety/neuroticism that is found within the Big
Five personality traits as emotional stability.
Also worth mentioning, the tendency to obsess over past experiences and stress is called "rumination":
State rumination, which involves dwelling on the consequences and
feelings associated with the failure. State rumination is more common
in people who are pessimistic, neurotic, and who have negative
attributional styles.
Rumination can also be measured as a personality trait using a variety of scales.