Any one elses babe going through seperation anxiety?

My girl is 10 months old. Such a good baby, but within the past couple weeks things have gotten very trying. I cannot leave her even a second without her screaming her head off. And often when I'm with her, I have to be paying full attention to her or she will also start screaming. She used to be good in public but now she screams everywhere wanting to get out of the cart/seat and explore. The only thing that calms her is puffs or her bottle but I'm tired of simply putting stuff in her mouth to calm her. Of course some of this off and on is teething, but I can tell when that's the case, but the seperation anxiety and newly found temper are the current issue. How long did it last for you? What do you do to help? At an older age, I would march her out of the store, but she's just a babe, so I'm not sure how to work on getting her to behave in public. Like what the first steps are? Am I supposed to keep her at my side and try to keep her happy, or should I be trying to let her cry it out and have play time on her own?