I know a lot of people love to run errands when their infant is asleep, but I actually always prefer to run errands when my kids are awake. I read this advice in a sleep book when I had my daughter (now 2), that kids sleep much better when there is no motion, and a well-rested child is a happy child. I followed that rule and made sleep a huge priority and my daughter has always been the happiest kid ever. She now actually asks for quiet time and nap time when she needs it. My son is two months old now, and he'll still usually fall asleep while I'm running errands, but I'm starting to look towards putting him on a sleep schedule in which we'll be home while he sleeps and out when he's awake. Plus this schedule makes life a little easier because the baby will sleep more when you're home and you'll be able to cook, clean, etc.
As for if you're worried about germs... with my daughter (first child and winter baby also) I actually made a paper sign with markers, laminated it, and pinned it to her or her carseat saying "Stop... please wash your hands before you touch mine." It was nicer than actually saying "don't touch the baby", but people got the hint, and EVERYONE said what a good idea it was (no one ever seemed offended or thought I was crazy). This time I'm not as worried (second time mom) so I haven't done it yet. I'm also more secure this time around just telling people not to touch him.