After you buy a few shipping lables online and successfully use them, you'll get over the worry. Soon you will find it hard to believe you ever did it the "old" way.
I think that after you buy a label, you are done. I don't think the post office will refund you any $. The difference between something that is 12 or 13 oz is minimal anyway, so don't worry about it. (FYI, I always round up, also. Each scale might be calibrated slightly different.)
NO, don't bother with going to the post office. Look to make your life easier. Once you've bought that label and attached it to your package, leave the package in your mailbox (if you can). If you are shipping priority, you can schedule a pick-up from your home/front door!