nah you're not allowing her to manipulate you, you are just having very normal guilt and overwhelming feelings while adjusting to parenting more than one child. I wish I could say it goes away at some point but honestly the guilt just changes.
As far as her behavior, 3 was tough. Very tough. We hardly left the house tough. Three and a half, ummm not much better. Four has been a very welcome change. Two year olds with their behavior ask the question, "how can I control me?", three year olds ask the question, "how can I control others?" and it's very normal for them to go through this stage. It's just your job as a parent to guide them on what is an appropriate way to channel their new found independence. I know right now with that teeny newborn in your arms she suddenly looks SOOOO big and grown but she's still so young and needs you & daddy for tons of help still.
If you have any time to read (honestly I know it's tough to find the time to pee right now) get a copy of Your Three Year Old by Ames & Ilg. You'll laugh and their hilarious accuracy in listing common behaviors for this age and you'll rest assured that she is quite well on the path of development for a three year old.