Landry was fired in February 1989. Jimmy Johnson drafted Aikman. I guess it’s possible that Landry discussed whether or not he wanted to draft Aikman, but I doubt it considering he was fired two months before the draft.
By a stroke of serendipity, Aikman and his UCLA Bruins were playing the Cotton Bowl in Dallas after the '88 season. Landry and Brandt were there; I even recall a picture of Landry with Aikman after the game (Cowboys Weekly probably). From all accounts, the big 3 (Schramm, Landry, and Brandt) were set on taking Aikman. They may have made some draft snafus in later years, but they weren't dumb. They did after all snag a one Michael Irvin a year earlier in the '88 draft, after Sherrard broke his fibula running on the beach and was never the same.
Aikman was gonna happen, even with the old regime. I'll never forget going into the last game of the 1988 season: both Dallas and Green Bay entered that week 3-12. GB won their game, Dallas lost. I was never more happy to see a Dallas loss, though even outside Aikman, there were some HOFers early in that draft (both Sanders). Too bad for GB they pick a dud in OT Tony Manderich! Then again, if GB got Aikman, then they probably don't trade for Favre in '91.