Yes, this is called neuroticism.

In personality psychology, anxiety, shame, and guilt, 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).

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.

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, the reason why all these emotions are subsumed under a single trait in the Big 5 is because they correlate so strongly (share an underlying factor). Personality inventories do not need to measure embarrassment specifically, as individuals high in embarrassability tend to score high in other neuroticism sub-facets (in fact, embarrassment scales are often validated against neuroticism).

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.

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.