Is it permissible to manually expel semen before going to the doctor to avoid sexual arousal during examination? And does that count as breaking the vow of not practicing masturbation if the intention is solely to avoid medical embarrassment?

You must inform the doctor of your condition as it is, without doing what you are asking about, even if it were permissible. How then, when it is forbidden, especially after your covenant with your Lord to abandon it? The hardship you face does not reach the level of necessity that permits the unlawful. The doctor is the most understanding of your excuse. Therefore, seek help from Allah and go to the doctor in the state you described.