Well, Mike- there were only a couple of HUNDRED different model shotguns sold by Sears! You could start by posting ALL the markings on your shotgun. For starters, does it say JC Higgins or Ted Williams? If you have a model number, I can maybe cross reference it to the maker and their original model number.
Sears has not actually MADE shotguns for about 85 years- they are made by someone else, with Sears name on them. For instance, I have a Sears JC Higgins Model 20 and a 21. They are High Standard Flite Kings- one a 12, other a 20.
Accuracy with a shotgun is a relative question- they shoot a pattern of shot. The bead front sight should be replaceable by any gunsmith for a few bucks. You may want to google search Blue Wonder for a DIY blue touch up. Bottom line- if it shoots at all, you got a good deal.
What we have here is... failure- to communicate.