Get out of your own head. No reason to be embarrassed. Stuttering is hard. Stop lying. Don't say you had a headache trying to conceal a truth that will come out. Stutterers are practiced at deception. We hide. We cheat. We avoid the direct path. I feel this is one of our greatest failings. I assure you, people finding out you lie about who you are will do greater harm to relationships than being a stutterer.