OK. Not strictly a Scottish band. But very much fronted by a singer from Scotland.
Here’s wiki.
The Tourists were a British rock and pop band. They achieved brief success in the late 1970s before the band split in 1980. Two of its members, singer Annie Lennox and guitarist Dave Stewart, went on to international success as Eurythmics.
Guitarists Peet Coombes and Dave Stewart were members of the folk rock band Longdancer which was on Elton John‘s Rocket Records label. They moved to London, where they met singer Annie Lennox, who had dropped out of a course at the Royal Academy of Music to pursue her ambitions in pop music.
Forming a band in 1976, the three of them initially called themselves The Catch. In 1977 the band released a single named “Borderline/Black Blood” on Logo Records. It was released in the UK, The Netherlands, Spain and Portugal, but was not a commercial success.
By 1976, they had recruited bass guitarist Eddie Chin and drummer Jim Toomey, and renamed themselves The Tourists. This saw the beginning of a productive period for the band and they released three albums: The Tourists (1979), Reality Effect (1979) and Luminous Basement (1980), as well as half a dozen singles, including “Blind Among the Flowers” (1979), “The Loneliest Man in the World” (1979), “Don’t Say I Told You So” (1980) and two hits, the Dusty Springfield cover “I Only Want to Be with You” (1979) and “So Good to Be Back Home Again” (1980), both of which reached the top 10 in the UK.
“I Only Want to Be with You” was also a top 10 hit in Australia and reached number 83 on the US Billboard Hot 100. Coombes was the band’s main songwriter, although later releases saw the first compositions by Lennox and Stewart.
In 1980, the band signed to RCA Records They toured extensively in the UK and abroad, including as support for Roxy Music on their 1979 Manifesto Tour. The group disbanded in late 1980.
Here’s one of the hit singles:-
mp3 : The Tourists – So Good To Be Back Home Again
I’ve been known to air this at the football when Raith Rovers haven’t played at Stark’s Park for a few weeks.
THE NEW VINYL VILLAIN 
The Tourists are perhaps understandably overshadowed by what came after and that’s rather a shame. The band seem to be a mere footnote in music of their time not really fitting into punk, post-punk, new wave or pop although they touched upon all of them.
I’d only ever think of The Tourists as Scottish – technically, I’d be wrong. Technically, I don’t care.
Even the much exploited androgeny was apparent at this time and with Eurythmics the band became pop – not always good pop – as they disappeared up their own arses with Touch.