Moffat Museum opened in the Old Moffat Bakehouse in 1984 and reopened in an extended building in 2013. Over two floors it covers local and family history: farming, trades and crafts, sport, schools, and Moffat's heyday as a spa, including the grand Hydropathic Hotel that burned down in 1921. There's also a model railway and the story of the town's long-gone branch line.
It's small and run by the community. Check the opening times locally.