My newborn daughter (born at 36 weeks 2 weeks old) has diarrhea, dry skin, sweating, runny nose temp of 97.8, and is a lot fussier. is this normal?

Yes: Depends upon how you are defining diarrhea here. The temp is normal. Sweating may or may not be ok. A runny nose usually isn't anything serious. If she is otherwise eating ok, i would watch it. If teh diarhea continues or she isn't eating normal then she needs to see your pediatrician.

M.D. visit: I agree that your daughter should be seen at her pediatrician's office. A newborn born slightly prematurely is susceptible to infection. If she has had diarrhea, more than 8 stools per day and has dry skin, she may be dehydrated. Sweating may indicate that she has a fever that has not been measured, and the fussiness can be associated with fever, pain, infection and a number of conditions.