I had the same problem you had, and this is what worked for me. Go to somewhere like CVS or Walgreens and get the saline nasal spray. Its usually a pretty small bottle, but its only water. The nosebleeds are happening because those membranes are getting all dries out in the nose. Shake it up good and give it a spray up each nasal in the morning, and then do it again at night before you go to bed, that will change everything. I usually pinched my nose after spraying just to get the liquid spread. The nosebleeds will stop. If you are going out shopping as you stated, maybe do a quick spray before going also as that will make sure it does occur while out.
You probably only need to do that in winter months when heaters are running everywhere and air is really dry.