Surprisingly, it appears that the odd Sears logo (with "sears" in a "fat," all-lowercase font) appeared in newspaper ads for quite some time. Specifically, that Sears logo was in use from approximately January 1, 1983 to June 30, 1984. During that time, that logo would often appear at the top of newspaper ads or circulars, while the familiar "Sears"-in-a-rectangle logo still appeared at the bottom of the ads.
Here's a 12/31/82 ad showing the rectangle logo at both the top and bottom of the page:
https://www.newspapers.com/article/john ... 144891047/
The first appearance of the fat, all-lowercase logo which I could find was from 1/1/83:
https://www.newspapers.com/article/sout ... 144891138/
And the last appearance of that logo which I could find was from 7/1/84:
https://www.newspapers.com/article/pres ... 144892290/
Interestingly, I found an ad from a few days prior (very similar to the one which the Groceteria Webmaster shared) about steaks that Sears was selling:
https://www.newspapers.com/article/pres ... 144891957/
The first appearance of the "racetrack" logo which I could find was from 6/30/84. Below are links to Sears ads from 6/30/84 and 7/1/84, as well as a link to a newspaper article explaining the logo change:
https://www.newspapers.com/article/the- ... 144891634/
https://www.newspapers.com/article/the- ... 144892451/
https://www.newspapers.com/article/asbu ... 144891792/
I am not sure why the fat, all-lowercase logo was used for 18 months. This is just speculation, but perhaps by the end of 1982, Sears felt that its rectangle logo was outdated but could not yet find a suitable replacement (hence the usage of a temporary logo which only appeared at the top of circulars and newspaper ads). If the newspaper article found by clicking on the link directly above is correct, apparently the process of redesigning of the Sears logo did not even begin until mid-1983, thus suggesting that the fat, all-lowercase logo was intended to be used only temporarily.