Have you checked the oil?, it may be very low and hot, which would cause the “ labored “ cranking. It may be possible that you have slow state of charge in your battery. A yearning lights before it died?
Have the new alternator checked out...flickering the headlights have nothing to do with the alternator...when you turn off the head lights just helps the alternator to send more voltage to the battery...when you give it gas it help the alternator increase the charge...you should not have to do all that...check you drive belt make sure is tight...alternator may need to be replace.
I take help from online websites that have this service like helping in repair some like r eliable-store have instant help.. others are a bit slow which I don't prefer... rest is up to u.. U can also get free manuals