A Chorus Lie - S3-E2
Factual error: A key plot point of this episode is the revelation of the Milli Vanilli lip-syncing scandal and that they didn't sing on their album, which happened in the fall of 1990, outside the 1980s setting of the series.
Suggested correction: The first public sign that the group was lip-syncing came on 21 July 1989, during a live performance on MTV at the Lake Compounce theme park in Bristol, Connecticut. As they performed, a hard drive issue caused the recording of the song "Girl You Know It's True" to jam and skip, repeatedly playing the partial line "Girl, you know it's..." through the speakers. "I knew right then and there, it was the beginning of the end for Milli Vanilli," recalled Pilatus of the incident.
The mistake didn't point it out, although the episode did, but the issue wasn't the fact that they lip-sync'd but the fact that they didn't actually do any singing. The minor incident you mentioned wasn't news that "shook the country." The news that it wasn't Milli Vanilli singing on their own album didn't come out until 1990.
My point exactly; while Milli Vanilli were revealed to be lip-syncing during a single live performance in the summer of 1989, it wasn't confirmed that they didn't sing at all on their debut album until November 1990.
I Caddyshacked the Pool - S3-E4
Factual error: We Are the World was new on TV (came out in 1985) but later Adam is seen playing a Gameboy, which was released in 1989.
Factual error: Whilst Adam and Erika discuss their grandfather buying a timeshare in Florida Adam states that their Grandfather is 80 years old. Erika responds that he was born in 1903 - thus setting the episode in 1983. However Adam is wearing a "Queen A Kind of Magic" T-shirt - an album that wasn't released until 1986.
Double Dare - S3-E13
Factual error: When watching a game, Barry mentions what the Eagles need to tie he mentions a two-point conversion. The NFL didn't adopt this until 1994, so it wouldn't have been an option in the 1980s.
Factual error: Older Adam says "33 years before Adele's Hello, there was Lionel Richie." Lionel Richie's Hello was released in 1984, Adele's version was 2015.