Let me say this as gently as I can. Your 6 month old baby, yes, baby, is adorable with this silliness. It might help you tremendously to do some reading on infant development. You have skipped from infant to terrible three's in your thinking here. And it's just going to make your life miserable if you don't figure that out and start enjoying her babyhood.
You don't have to hold her all the time. You don't have to look at her all the time. But you cannot stop her from crying when you leave the room. You SHOULD NOT even be thinking of stopping the Muuuummm either. You can move her from room to room with you while you work, taking breaks to have floor/tummy time with her to help build her muscles and bond with her. You can sing to her, talk to her while you do your work. That's what people do with little babies. A six month old does not yet even have the capacity to understand that when you leave the room, that you will come back. She is totally dependent on you and when you walk out, she thinks you disappeared for good. She has to develop into the understanding that you will come back. Your repeated overatures to her that you are still there for her, by talking sweetly, singing, holding her some, moving her around with you, meeting her physical needs in every way as well, will develop trust and security for her that will make the person that she will end up being.
You cannot control which person she would rather be with, you or Daddy. It's not your call. Some babies are totally attached to mom. Some treat mom like chopped liver when dad walks through the door at the end of the day and then will have nothing to do with her until he leaves for work the next day. You don't get to pick your child's personality, though you can help shape it. You and your husband continue to be good parents through it all, and understand that babies are just that way.
Your other child is the older one. Let the baby be a baby. She will change so fast from month to month that your head will spin. Enjoy this phase, accept this phase, and don't make her be old in your mind. You'll look back on this time and smile in your heart.
Dawn