Going to a January 1986 calendar and crossing out the dates of the games that
are in the Archive, and also the dates a day before and a day after (given that the three games in question did not have three new players on the first among them and did not end with no returning champion), and the weekends, of course, leaves fairly limited possibilities.
Assuming the three games are consecutive (they are) and all aired in January (who knows?), they either aired
Jan. 1-3 (Wednesday-Friday, taped September 4, 1985),
Jan. 2-6 (Thursday-Monday, taped September 4 and 9, 1985),
Jan. 16-20 (Thursday-Monday, taped September 10 and 17, 1985),
Jan. 17-21 (Friday-Tuesday, taped September 10 and 17, 1985), or
Jan. 20-22 (Monday-Wednesday, taped September 17, 1985).
Getting a hint as to the weekday of any particular game could narrow it down further.
Unfortunately, Alex does not bid the audience "see you tomorrow" or "see you Monday" in any of the three games, but instead makes reference to a vague "next time" in each of the three closings.
The games are:
Bill Fleming (1xer with 11,400) vs. Joe (Jose) Escarce vs. Linda Shrum
Bill Fleming (2xer) vs. Linda Shrum (1xer with $8,600) vs. Geoff Fong (2nd time in just the past week and a half there have been co-champions, 3rd or 4th time this "year" (season or month?))
Linda Shrum (2xer with $17,700) vs. Rosalie Hill (winner of this game with $9,750) vs. Kevin Spitze
Here is
Rosalie Hill's existing player page in the J! Archive. I am able to fill in the "Season 2 X-time champion" to make it "Season 2 1-time champion".
#351, the game aired Monday, January 13, 1986 and taped September 10, 1985, has co-champions. That
may not completely eliminate the Jan. 1-3 and 2-6 possibilities, above, but it makes them less likely.
Based on Alex's "last week and a half" language, I think January 20-22, 1986 is the best bet. Good enough for Archive entry with an "airdate approximate" note.