Nonstop comfort nursing

My baby is 5 weeks old and she has been nonstop nursing. If she is awake she is fussy and has gas and is constantly rooting/wanting to nurse for comfort. She is rarely ever content when awake unless she is nursing. She will not take a pacifier. I have just about every brand possible and have tried everything to get her to take one.

I feel trapped. I had planned to stop breastfeeding at 2 months but I feel I will have no way to console her if I quit. She will purple cry and not give up until I give in. I have tried rocking her, walking, singing, music. Nothing works except nursing :( I know many will say growth spurts or cluster feeding but she is not swallowing most of the time. Will she grow out of this?