Coping with demanding, tantrum throwing 13 month old!?

My 4th child, Josie is quite the handful. She is smart as a whip, but oh-so prone to emotional meltdowns! She is doted on by her siblings, and gets plenty of attention. She has to have me and only me, all day every day. She's not keen on Dad or any of the grandparents. Dad tries hard to be her buddy. She likes her 8 year old sister and 4 year old brother, but is mean to her 15 yr. old sister. She's screams and "yells" at her brother if he has anything she happens to want, and she'll cry until she's blue in the face or her lungs are about to explode about anything in which she doesn't get her way. Everything i have read concerning tantrums pertains to older toddlers that you can talk to and reason with. She is just barely over one year old. Granted, she did crawl early (6 mo.), and she was walking by 10 mo., but i do not know what to do! She's too young to put in a corner for timeout or . We've tried to ignore her, but as i've said, she'll go comatose screaming just to get a reaction! My husband is also the short-fused, short tempered type, and he is fit to be tied with her. Even though i do almost 99% of her care, if he's home while she's screaming, i have to hear it from both ends. I kind of think her problem is that she is too much like her dad, but i also feel that she is too young. Other times, she is the sweetest, most loving, kid. I truly feel as if i have a Dr. Jekyll and Mr. Hyde on my hands. In a nutshell: if she were the child that we had first, there would be no others! My other three are alot more mild mannered and i never had to deal with any tantrums before. Help! She's almost driving me to drink!!! LOL!