Good morning, K.,
I know this is a really LONG response; sometimes I wish we could voice record our advice, so it would be faster!
I don't know how you feel about slings, but that was my life saver with my daughter. She was also very much a "mama's girl". Because I had a three year old son (and ran a home daycare) I could not deal with her crying pathetically while I just went about my day. I figured, if she feels she needs me, then she needs me - why deny her that? I didn't want her to be an insecure, clingy child for the rest of her life, so I gave her what she needed - ME! I wore her (what seemed like) constantly in the sling, for the first two years of her life. That left me hands free to do what I needed with the other children, it gave us a peaceful day, because she was utterly content, and it made it easy to feed her while she was still nursing. You'll notice I said "what seemed like". That's because, of course, it WASN'T constantly. Once she had the needed "mama time" she felt secure enough to get down and play for a while. And as she got older, those "whiles" became longer and longer, until she finally didn't feel the need to be in the sling.
I know that there are plenty of moms out there who are rolling their eyes at my response. But I have the proof that it worked. My daughter is now a very happy, independent, well adjusted six year old.
So, if your daughter is not big for her age, and you have a sling, that is my advice.
If you cannot do that, then I would advise you to pick her up when she needs it and go about your day. For example, she's awake now, and she only wants YOU to get her up, but you need to tend to your 3 yo. Go in, hug her for a minute, then say (with enthusiasm) "Okay, sweet girl, mama has to get breakfast ready and I need you to help me! Come on! I need help. You can help me pour the cereal!" Then let her "help" you get her sister's breakfast. Children LOVE to help, if they feel you really need it.
Continue this pattern, -- "okay, now we have to pick out some clothes to help sister get dressed, what do you think we should pick out?" Try to get her to walk with you, but if she balks, don't make a big deal out of it - just pick her up and take her along. "There; now we have to sort the laundry. Come on, we have lots of work to do and I need help." Each time you transition, try to get her to do it without you carrying her. Eventually she will want to move away from you and do something else that has caught her interest. And that will get more frequent and for longer periods, as the weeks pass.
Best of luck to you, however you decide to handle it. Hopefully it will get resolved sooner rather than later!
Blessings,
J.