LT still has his MVP season, it just comes in the strike-shortened 1987 season, as a member of the Saints.
It's also the first year of this revamped "Dome Patrol" linebacker corps, with Taylor taking the place of Rickey Jackson, who the Saints also drafted in 1981. For the sake of simplicity, we'll just say the Giants end up with Rogers and Jackson in this universe, so they are still a very good team, just not a champion.
Jackson is in Canton after all.
As for the rest of the 1987 season for the Saints, with MVP LT leading the historically great defense, the Saints don't take a nap against the Minnesota Vikings in the first round, get past their division rival San Francisco 49ers in the divisional round (a team LT's Giants often had success against) before defeating Washington in the NFC Championship, and advancing to Super Bowl XXII. Here though, they fall to Elway's Broncos due to being unable to sustain much offensively thanks to limited firepower.
Remaining contenders the next couple years thanks to momentum, the Saints have to wait a bit before getting back to the dance. Fear not though Saints fans.
With that said, we now travel to the 1990 season. Leading up to this year, the Saints decide they need to build their offense to go with their all-world defense. In 1988, they still draft Craig "Ironhead" Heyward, but in 1989, partially due to having a later pick, they draft wide receiver Andre Rison, instead of defensive lineman Wayne Martin, and all of a sudden, the offense is looking a lot more dandy on paper.
Then there's the quarterback.