Help! My 5 week old won't let me put her down.

My baby is 5 weeks old and is only content in any sort of swing/bouncy seat/bassinet for 5-10 minutes. After these few minutes, she gets upset and cries. This occurs all day and night, also making it impossible for her to sleep without being on my chest. I really don't want to use the cry it out method but I don't know what else to do. I am terrified when she goes to daycare in the next few weeks that the provider won't be able to take care of the other kids leaving me without daycare. Please help!