The L&P rep was no doubt talking about the sauce made in Worcester, England. Every batch I tried over the last 35 years tasted different from the sauce made by L&P in Fair Lawn, NJ. However, I usually use Baumer Foods’ Worcestershire sauce, sold either under their own Crystal name or under a million house brands. It goes off after about 4 years from time of manufacture.
If there is sludge at the bottom of your bottle that shaking will not break up, I would say it might not be spoiled, but would change the flavor of the dish.