I will now outline how to deal with this type of person whilst in the store:
1) Aisle the first: since she will see you first and alarm you with her presence, you will be caught off-guard to the point of being unable to recall her name until long after you've left the store. So a politely awkward "Oh! Hello...there!" will do just fine. Smile brightly as you quickly move on from one another to continue your shopping.
2) Aisle the Second: you run into Lady X yet again, and you realize with growing horror that you both are at the beginning of your shopping experience, you both move at an equal pace, and you're both shopping in the same aisles at the store. You now know that you will be seeing her in every aisle for the rest of your shopping day.
3) Skip aisles three and four in an attempt to get in a different pattern from your new shopping mate, only to note later that she's had the same brilliant idea.
4) Race through the dairy section to get ahead of her, only to nearly run over her offspring upon exiting at the yogurt wall.
5) Aisle the Fifth: Quickly formulate a reason to speak to her since it's becoming obvious you cannot avoid her to save your life, i.e., think of something to say to her, stupid. Again smile brightly, act normal for once, and say, "Soooooo...is your husband in the Army?" She seems delighted that you've saved the day, however briefly, by stabbing at the engulfing awkwardness with some attempt at regular conversation. "No, Air Force." "Oh, I see." 30 seconds of silence ensue while you realize that you failed to think of any follow-up questions to your newly invented conversation. You ask a few more fool questions before both of you tire of the made-up attempt at being friendly and you just sort of aimlessly walk away from her as she gratefully runs in the other direction.
6) See her scoot down Aisle Six, and slyly move to Aisle Seven, hoping to backtrack after she's left. When you go to turn against traffic to go to Aisle Six, be sure to ram into her cart going in the opposite direction. Apologize profusely while she assures you she's fine. Be half grateful you managed to find another excuse to say something a normal person would say, despite the fact it's becoming increasingly obvious to this woman that you're anything but normal.
7) Run to Meats, turn to see her coming, and grab as many random packaged beef products you can before she overtakes you. Run away.
8) Confident that you're now ahead of her, take your merry time down the bread aisle and make for your getaway at check-out, but not before yet again, nearly running into her on your way out. Smile ever so briefly while making as little eye-contact as possible. Her own motivation to fake a smile has vanished, and she pretends she no longer knows you. Totally fine with you.
9) Check out and run to your car. Pray for death before next Sunday.